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+
+
+ + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/api/eigensolvers/index.html b/api/eigensolvers/index.html new file mode 100644 index 0000000..38ea24e --- /dev/null +++ b/api/eigensolvers/index.html @@ -0,0 +1,1189 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + eigensolvers - meanas + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + Skip to content + + +
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+ + + +
+ +
+ + + + + +

eigensolvers

+ + +
+ + + +

+ meanas.eigensolvers + + +

+ +
+ +

Solvers for eigenvalue / eigenvector problems

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ power_iteration + + +

+
power_iteration(
+    operator: spmatrix,
+    guess_vector: NDArray[complex128] | None = None,
+    iterations: int = 20,
+) -> tuple[complex, NDArray[numpy.complex128]]
+
+ +
+ +

Use power iteration to estimate the dominant eigenvector of a matrix.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ operator + + spmatrix + +
+

Matrix to analyze.

+
+ 		+ required +
+ guess_vector + + NDArray[complex128] | None + +
+

Starting point for the eigenvector. Default is a randomly chosen vector.

+
+ 		+ None +
+ iterations + + int + +
+

Number of iterations to perform. Default 20.

+
+ 		+ 20 +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[complex, NDArray[complex128]] + +
+

(Largest-magnitude eigenvalue, Corresponding eigenvector estimate)

+
+
+ + +
+ +
+ +
+ + +

+ rayleigh_quotient_iteration + + +

+
rayleigh_quotient_iteration(
+    operator: spmatrix | LinearOperator,
+    guess_vector: NDArray[complex128],
+    iterations: int = 40,
+    tolerance: float = 1e-13,
+    solver: Callable[..., NDArray[complex128]]
+    | None = None,
+) -> tuple[complex, NDArray[numpy.complex128]]
+
+ +
+ +

Use Rayleigh quotient iteration to refine an eigenvector guess.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ operator + + spmatrix | LinearOperator + +
+

Matrix to analyze.

+
+ 		+ required +
+ guess_vector + + NDArray[complex128] + +
+

Eigenvector to refine.

+
+ 		+ required +
+ iterations + + int + +
+

Maximum number of iterations to perform. Default 40.

+
+ 		+ 40 +
+ tolerance + + float + +
+

Stop iteration if (A - I*eigenvalue) @ v < num_vectors * tolerance, + Default 1e-13.

+
+ 		+ 1e-13 +
+ solver + + Callable[..., NDArray[complex128]] | None + +
+

Solver function of the form x = solver(A, b). + By default, use scipy.sparse.spsolve for sparse matrices and + scipy.sparse.bicgstab for general LinearOperator instances.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[complex, NDArray[complex128]] + +
+

(eigenvalues, eigenvectors)

+
+
+ + +
+ +
+ +
+ + +

+ signed_eigensolve + + +

+
signed_eigensolve(
+    operator: spmatrix | LinearOperator,
+    how_many: int,
+    negative: bool = False,
+) -> tuple[
+    NDArray[numpy.complex128], NDArray[numpy.complex128]
+]
+
+ +
+ +

Find the largest-magnitude positive-only (or negative-only) eigenvalues and + eigenvectors of the provided matrix.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ operator + + spmatrix | LinearOperator + +
+

Matrix to analyze.

+
+ 		+ required +
+ how_many + + int + +
+

How many eigenvalues to find.

+
+ 		+ required +
+ negative + + bool + +
+

Whether to find negative-only eigenvalues. + Default False (positive only).

+
+ 		+ False +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ NDArray[complex128] + +
+

(sorted list of eigenvalues, 2D ndarray of corresponding eigenvectors)

+
+
+ NDArray[complex128] + +
+

eigenvectors[:, k] corresponds to the k-th eigenvalue

+
+
+ + +
+ +
+ + + +
+ +
+ +
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+
+ + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/api/fdfd/index.html b/api/fdfd/index.html new file mode 100644 index 0000000..07b78ef --- /dev/null +++ b/api/fdfd/index.html @@ -0,0 +1,5211 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + fdfd - meanas + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + Skip to content + + +
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+ +
+ + + + + +

fdfd

+ + +
+ + + +

+ meanas.fdfd + + +

+ +
+ +

Tools for finite difference frequency-domain (FDFD) simulations and calculations.

+

These mostly involve picking a single frequency, then setting up and solving a +matrix equation (Ax=b) or eigenvalue problem.

+

Submodules:

+
    +
  • operators, functional: General FDFD problem setup.
    • +
  • solvers: Solver interface and reference implementation.
    • +
  • scpml: Stretched-coordinate perfectly matched layer (SCPML) boundary conditions.
    • +
  • waveguide_2d: Operators and mode-solver for waveguides with constant cross-section.
    • +
  • waveguide_3d: Functions for transforming waveguide_2d results into 3D, + including mode-source and overlap-window construction.
    • +
  • farfield, bloch, eme: specialized helper modules for near/far transforms, + Bloch-periodic problems, and eigenmode expansion.
    • +
+

================================================================

+

From the "Frequency domain" section of meanas.fdmath, we have

+
\[ + \begin{aligned} + \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\ + \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\ + \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\ + \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}}) + -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\ + \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t + \end{aligned} +\]
+

resulting in

+
\[ + \begin{aligned} + \tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\ + \hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\ + \end{aligned} +\]
+

Maxwell's equations are then

+
\[ + \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{\vec{r}} &= + \imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}} + - \hat{M}_{\vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &= + -\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}} + + \tilde{J}_{\vec{r}} \\ + \tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\ + \hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}} + \end{aligned} +\]
+

With \(\Delta_t \to 0\), this simplifies to

+
\[ + \begin{aligned} + \tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\ + \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\ + \tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\ + \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\ + \Omega &\to \omega \\ + \tilde{\partial}_t &\to -\imath \omega \\ + \hat{\partial}_t &\to -\imath \omega \\ + \end{aligned} +\]
+

and then

+
\[ + \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{\vec{r}} &= + \imath \omega \hat{B}_{\vec{r} + \frac{1}{2}} + - \hat{M}_{\vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &= + -\imath \omega \tilde{D}_{\vec{r}} + + \tilde{J}_{\vec{r}} \\ + \end{aligned} +\]
+
\[ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}}) + -\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\ +\]
+ + + + + + + + + + +
+ + + + + + + + + + + + +
+ +
+ +

Core operator layers

+ + +
+ + + +

+ meanas.fdfd.functional + + +

+ +
+ +

Functional versions of many FDFD operators. These can be useful for performing + FDFD calculations without needing to construct large matrices in memory.

+

The functions generated here expect cfdfield_t inputs with shape (3, X, Y, Z), +e.g. E = [E_x, E_y, E_z] where each (complex) component has shape (X, Y, Z)

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ e_full + + +

+
e_full(
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: fdfield,
+    mu: fdfield | None = None,
+) -> cfdfield_updater_t
+
+ +
+ +

Wave operator for use with E-field. See operators.e_full for details.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + fdfield + +
+

Dielectric constant

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ cfdfield_updater_t + +
+

Function f implementing the wave operator

+
+
+ cfdfield_updater_t + +
+

f(E) -> -i * omega * J

+
+
+ + +
+ +
+ +
+ + +

+ eh_full + + +

+
eh_full(
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: fdfield,
+    mu: fdfield | None = None,
+) -> Callable[
+    [cfdfield, cfdfield], tuple[cfdfield_t, cfdfield_t]
+]
+
+ +
+ +

Wave operator for full (both E and H) field representation. +See operators.eh_full.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + fdfield + +
+

Dielectric constant

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ Callable[[cfdfield, cfdfield], tuple[cfdfield_t, cfdfield_t]] + +
+

Function f implementing the wave operator

+
+
+ Callable[[cfdfield, cfdfield], tuple[cfdfield_t, cfdfield_t]] + +
+

f(E, H) -> (J, -M)

+
+
+ + +
+ +
+ +
+ + +

+ e2h + + +

+
e2h(
+    omega: complex,
+    dxes: dx_lists_t,
+    mu: fdfield | None = None,
+) -> cfdfield_updater_t
+
+ +
+ +

Utility operator for converting the E field into the H field. +For use with e_full -- assumes that there is no magnetic current M.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ cfdfield_updater_t + +
+

Function f for converting E to H,

+
+
+ cfdfield_updater_t + +
+

f(E) -> H

+
+
+ + +
+ +
+ +
+ + +

+ m2j + + +

+
m2j(
+    omega: complex,
+    dxes: dx_lists_t,
+    mu: fdfield | None = None,
+) -> cfdfield_updater_t
+
+ +
+ +

Utility operator for converting magnetic current M distribution +into equivalent electric current distribution J. +For use with e.g. e_full.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ cfdfield_updater_t + +
+

Function f for converting M to J,

+
+
+ cfdfield_updater_t + +
+

f(M) -> J

+
+
+ + +
+ +
+ +
+ + +

+ e_tfsf_source + + +

+
e_tfsf_source(
+    TF_region: fdfield,
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: fdfield,
+    mu: fdfield | None = None,
+) -> cfdfield_updater_t
+
+ +
+ +

Operator that turns an E-field distribution into a total-field/scattered-field +(TFSF) source.

+

If A is the full wave operator from e_full(...) and Q is the diagonal +mask selecting the total-field region, then the TFSF source is the commutator

+
\[ +\frac{A Q - Q A}{-i \omega} E. +\]
+

This vanishes in the interior of the total-field and scattered-field regions +and is supported only at their shared boundary, where the mask discontinuity +makes A and Q fail to commute. The returned current is therefore the +distributed source needed to inject the desired total field without also +forcing the scattered-field region.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ TF_region + + fdfield + +
+

mask which is set to 1 in the total-field region, and 0 elsewhere + (i.e. in the scattered-field region). + Should have the same shape as the simulation grid, e.g. epsilon[0].shape.

+
+ 		+ required +
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + fdfield + +
+

Dielectric constant distribution

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ cfdfield_updater_t + +
+

Function f which takes an E field and returns a current distribution,

+
+
+ cfdfield_updater_t + +
+

f(E) -> J

+
+
+ + +
+ +
+ +
+ + +

+ poynting_e_cross_h + + +

+
poynting_e_cross_h(
+    dxes: dx_lists_t,
+) -> Callable[[cfdfield, cfdfield], cfdfield_t]
+
+ +
+ +

Generates a function that takes the single-frequency E and H fields +and calculates the cross product E x H = \(E \times H\) as required +for the Poynting vector, \(S = E \times H\).

+

On the Yee grid, the electric and magnetic components are not stored at the +same locations. This helper therefore applies the same one-cell electric-field +shifts used by the sparse operators.poynting_e_cross(...) construction so +that the discrete cross product matches the face-centered energy flux used in +meanas.fdtd.energy.poynting(...).

+ + +
+ Note +

This function also shifts the input E field by one cell as required +for computing the Poynting cross product (see meanas.fdfd module docs).

+
+ +
+ Note +

If E and H are peak amplitudes as assumed elsewhere in this code, +the time-average of the poynting vector is <S> = Re(S)/2 = Re(E x H*) / 2. +The factor of 1/2 can be omitted if root-mean-square quantities are used +instead.

+
+ +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ Callable[[cfdfield, cfdfield], cfdfield_t] + +
+

Function f that returns the staggered-grid cross product E \times H.

+
+
+ Callable[[cfdfield, cfdfield], cfdfield_t] + +
+

For time-average power, call it as f(E, H.conj()) and take Re(...) / 2.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdfd.operators + + +

+ +
+ +

Sparse matrix operators for use with electromagnetic wave equations.

+

These functions return sparse-matrix (scipy.sparse.sparray) representations of + a variety of operators, intended for use with E and H fields vectorized using the + meanas.fdmath.vectorization.vec() and meanas.fdmath.vectorization.unvec() functions.

+

E- and H-field values are defined on a Yee cell; epsilon values should be calculated for + cells centered at each E component (mu at each H component).

+

Many of these functions require a dxes parameter, of type dx_lists_t; see +the meanas.fdmath.types submodule for details.

+

The following operators are included:

+
    +
  • E-only wave operator
    • +
  • H-only wave operator
    • +
  • EH wave operator
    • +
  • Curl for use with E, H fields
    • +
  • E to H conversion
    • +
  • M to J conversion
    • +
  • Poynting cross products
    • +
  • Circular shifts
    • +
  • Discrete derivatives
    • +
  • Averaging operators
    • +
  • Cross product matrices
    • +
+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ e_full + + +

+
e_full(
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: vfdfield | vcfdfield,
+    mu: vfdfield | None = None,
+    pec: vfdfield | None = None,
+    pmc: vfdfield | None = None,
+) -> sparse.sparray
+
+ +
+ +

Wave operator + $$ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon $$

+
del x (1/mu * del x) - omega**2 * epsilon
+
+

for use with the E-field, with wave equation + $$ (\nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon) E = -\imath \omega J $$

+
(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
+
+

To make this matrix symmetric, use the preconditioners from e_full_preconditioners().

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + vfdfield | vcfdfield + +
+

Vectorized dielectric constant

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere).

+
+ 		+ None +
+ pec + + vfdfield | None + +
+

Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted +as containing a perfect electrical conductor (PEC). +The PEC is applied per-field-component (i.e. pec.size == epsilon.size)

+
+ 		+ None +
+ pmc + + vfdfield | None + +
+

Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted +as containing a perfect magnetic conductor (PMC). +The PMC is applied per-field-component (i.e. pmc.size == epsilon.size)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix containing the wave operator.

+
+
+ + +
+ +
+ +
+ + +

+ e_full_preconditioners + + +

+
e_full_preconditioners(
+    dxes: dx_lists_t,
+) -> tuple[sparse.sparray, sparse.sparray]
+
+ +
+ +

Left and right preconditioners (Pl, Pr) for symmetrizing the e_full wave operator.

+

The preconditioned matrix A_symm = (Pl @ A @ Pr) is complex-symmetric + (non-Hermitian unless there is no loss or PMLs).

+

The preconditioner matrices are diagonal and complex, with Pr = 1 / Pl

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[sparray, sparray] + +
+

Preconditioner matrices (Pl, Pr).

+
+
+ + +
+ +
+ +
+ + +

+ h_full + + +

+
h_full(
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: vfdfield,
+    mu: vfdfield | None = None,
+    pec: vfdfield | None = None,
+    pmc: vfdfield | None = None,
+) -> sparse.sparray
+
+ +
+ +

Wave operator + $$ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu $$

+
del x (1/epsilon * del x) - omega**2 * mu
+
+

for use with the H-field, with wave equation + $$ (\nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu) E = \imath \omega M $$

+
(del x (1/epsilon * del x) - omega**2 * mu) E = i * omega * M
+
+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + vfdfield + +
+

Vectorized dielectric constant

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ pec + + vfdfield | None + +
+

Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted +as containing a perfect electrical conductor (PEC). +The PEC is applied per-field-component (i.e. pec.size == epsilon.size)

+
+ 		+ None +
+ pmc + + vfdfield | None + +
+

Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted +as containing a perfect magnetic conductor (PMC). +The PMC is applied per-field-component (i.e. pmc.size == epsilon.size)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix containing the wave operator.

+
+
+ + +
+ +
+ +
+ + +

+ eh_full + + +

+
eh_full(
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: vfdfield,
+    mu: vfdfield | None = None,
+    pec: vfdfield | None = None,
+    pmc: vfdfield | None = None,
+) -> sparse.sparray
+
+ +
+ +

Wave operator for [E, H] field representation. This operator implements Maxwell's + equations without cancelling out either E or H. The operator is +$$ \begin{bmatrix} + -\imath \omega \epsilon & \nabla \times \ + \nabla \times & \imath \omega \mu + \end{bmatrix} $$

+
[[-i * omega * epsilon,  del x         ],
+ [del x,                 i * omega * mu]]
+
+

for use with a field vector of the form cat(vec(E), vec(H)): +$$ \begin{bmatrix} + -\imath \omega \epsilon & \nabla \times \ + \nabla \times & \imath \omega \mu + \end{bmatrix} + \begin{bmatrix} E \ + H + \end{bmatrix} + = \begin{bmatrix} J \ + -M + \end{bmatrix} $$

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + vfdfield + +
+

Vectorized dielectric constant

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ pec + + vfdfield | None + +
+

Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted +as containing a perfect electrical conductor (PEC). +The PEC is applied per-field-component (i.e. pec.size == epsilon.size)

+
+ 		+ None +
+ pmc + + vfdfield | None + +
+

Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted +as containing a perfect magnetic conductor (PMC). +The PMC is applied per-field-component (i.e. pmc.size == epsilon.size)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix containing the wave operator.

+
+
+ + +
+ +
+ +
+ + +

+ e2h + + +

+
e2h(
+    omega: complex,
+    dxes: dx_lists_t,
+    mu: vfdfield | None = None,
+    pmc: vfdfield | None = None,
+) -> sparse.sparray
+
+ +
+ +

Utility operator for converting the E field into the H field. +For use with e_full() -- assumes that there is no magnetic current M.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ pmc + + vfdfield | None + +
+

Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted +as containing a perfect magnetic conductor (PMC). +The PMC is applied per-field-component (i.e. pmc.size == epsilon.size)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for converting E to H.

+
+
+ + +
+ +
+ +
+ + +

+ m2j + + +

+
m2j(
+    omega: complex,
+    dxes: dx_lists_t,
+    mu: vfdfield | None = None,
+) -> sparse.sparray
+
+ +
+ +

Operator for converting a magnetic current M into an electric current J. +For use with eg. e_full().

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for converting M to J.

+
+
+ + +
+ +
+ +
+ + +

+ poynting_e_cross + + +

+
poynting_e_cross(
+    e: vcfdfield, dxes: dx_lists_t
+) -> sparse.sparray
+
+ +
+ +

Operator for computing the staggered-grid (E \times) part of the Poynting vector.

+

On the Yee grid the E and H components live on different edges, so the +electric field must be shifted by one cell in the appropriate direction +before forming the discrete cross product. This sparse operator encodes that +shifted cross product directly and is the matrix equivalent of +functional.poynting_e_cross_h(...).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e + + vcfdfield + +
+

Vectorized E-field for the ExH cross product

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix containing the (E \times) part of the staggered Poynting

+
+
+ sparray + +
+

cross product.

+
+
+ + +
+ +
+ +
+ + +

+ poynting_h_cross + + +

+
poynting_h_cross(
+    h: vcfdfield, dxes: dx_lists_t
+) -> sparse.sparray
+
+ +
+ +

Operator for computing the staggered-grid (H \times) part of the Poynting vector.

+

Together with poynting_e_cross(...), this provides the matrix form of the +Yee-grid cross product used in the flux helpers. The two are related by the +usual antisymmetry of the cross product,

+
\[ +H \times E = -(E \times H), +\]
+

once the same staggered field placement is used on both sides.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ h + + vcfdfield + +
+

Vectorized H-field for the HxE cross product

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix containing the (H \times) part of the staggered Poynting

+
+
+ sparray + +
+

cross product.

+
+
+ + +
+ +
+ +
+ + +

+ e_tfsf_source + + +

+
e_tfsf_source(
+    TF_region: vfdfield,
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: vfdfield,
+    mu: vfdfield | None = None,
+) -> sparse.sparray
+
+ +
+ +

Operator that turns a desired E-field distribution into a + total-field/scattered-field (TFSF) source.

+

Let A be the full wave operator from e_full(...), and let +Q = \mathrm{diag}(TF_region) be the projector onto the total-field region. +Then the TFSF current operator is the commutator

+
\[ +\frac{A Q - Q A}{-i \omega}. +\]
+

Inside regions where Q is locally constant, A and Q commute and the +source vanishes. Only cells at the TF/SF boundary contribute nonzero current, +which is exactly the desired distributed source for injecting the chosen +field into the total-field region without directly forcing the +scattered-field region.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ TF_region + + vfdfield + +
+

Mask, which is set to 1 inside the total-field region and 0 in the + scattered-field region

+
+ 		+ required +
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + vfdfield + +
+

Vectorized dielectric constant

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere).

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix that turns an E-field into a current (J) distribution.

+
+
+ + +
+ +
+ +
+ + +

+ e_boundary_source + + +

+
e_boundary_source(
+    mask: vfdfield,
+    omega: complex,
+    dxes: dx_lists_t,
+    epsilon: vfdfield,
+    mu: vfdfield | None = None,
+    periodic_mask_edges: bool = False,
+) -> sparse.sparray
+
+ +
+ +

Operator that turns an E-field distrubtion into a current (J) distribution + along the edges (external and internal) of the provided mask. This is just an + e_tfsf_source() with an additional masking step.

+

Equivalently, this helper first constructs the TFSF commutator source for the +full mask and then zeroes out all cells except the mask boundary. The +boundary is defined as the set of cells whose mask value changes under a +one-cell shift in any Cartesian direction. With periodic_mask_edges=False +the shifts mirror at the domain boundary; with True they wrap periodically.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ mask + + vfdfield + +
+

The current distribution is generated at the edges of the mask, + i.e. any points where shifting the mask by one cell in any direction + would change its value.

+
+ 		+ required +
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + vfdfield + +
+

Vectorized dielectric constant

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Vectorized magnetic permeability (default 1 everywhere).

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix that turns an E-field into a current (J) distribution.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdfd.solvers + + +

+ +
+ +

Solvers and solver interface for FDFD problems.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ generic + + +

+
generic(
+    omega: complex,
+    dxes: dx_lists_t,
+    J: vcfdfield,
+    epsilon: vfdfield,
+    mu: vfdfield | None = None,
+    *,
+    pec: vfdfield | None = None,
+    pmc: vfdfield | None = None,
+    adjoint: bool = False,
+    matrix_solver: Callable[..., ArrayLike] = _scipy_qmr,
+    matrix_solver_opts: dict[str, Any] | None = None,
+    E_guess: vcfdfield | None = None,
+) -> vcfdfield_t
+
+ +
+ +

Conjugate gradient FDFD solver using CSR sparse matrices.

+

All ndarray arguments should be 1D arrays, as returned by meanas.fdmath.vectorization.vec().

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

Complex frequency to solve at.

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

[[dx_e, dy_e, dz_e], [dx_h, dy_h, dz_h]] (complex cell sizes) as +discussed in meanas.fdmath.types

+
+ 		+ required +
+ J + + vcfdfield + +
+

Electric current distribution (at E-field locations)

+
+ 		+ required +
+ epsilon + + vfdfield + +
+

Dielectric constant distribution (at E-field locations)

+
+ 		+ required +
+ mu + + vfdfield | None + +
+

Magnetic permeability distribution (at H-field locations)

+
+ 		+ None +
+ pec + + vfdfield | None + +
+

Perfect electric conductor distribution + (at E-field locations; non-zero value indicates PEC is present)

+
+ 		+ None +
+ pmc + + vfdfield | None + +
+

Perfect magnetic conductor distribution + (at H-field locations; non-zero value indicates PMC is present)

+
+ 		+ None +
+ adjoint + + bool + +
+

If true, solves the adjoint problem.

+
+ 		+ False +
+ matrix_solver + + Callable[..., ArrayLike] + +
+

Called as matrix_solver(A, b, **matrix_solver_opts) -> x, + where A: scipy.sparse.csr_array; + b: ArrayLike; + x: ArrayLike; + Default is a wrapped version of scipy.sparse.linalg.qmr() + which doesn't return convergence info and logs the residual + every 100 iterations.

+
+ 		+ _scipy_qmr +
+ matrix_solver_opts + + dict[str, Any] | None + +
+

Passed as kwargs to matrix_solver(...)

+
+ 		+ None +
+ E_guess + + vcfdfield | None + +
+

Guess at the solution E-field. matrix_solver must accept an + x0 argument with the same purpose.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ vcfdfield_t + +
+

E-field which solves the system.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdfd.scpml + + +

+ +
+ +

Functions for creating stretched coordinate perfectly matched layer (PML) absorbers.

+ + + + + + + + + + +
+ + + + + + + +
+ + + +

+ s_function_t + + + + module-attribute + + +

+
s_function_t = Callable[
+    [NDArray[float64]], NDArray[float64]
+]
+
+ +
+ +

Typedef for s-functions, see prepare_s_function()

+ +
+ +
+ + + + +
+ + +

+ prepare_s_function + + +

+
prepare_s_function(
+    ln_R: float = -16, m: float = 4
+) -> s_function_t
+
+ +
+ +

Create an s_function to pass to the SCPML functions. This is used when you would like to +customize the PML parameters.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ ln_R + + float + +
+

Natural logarithm of the desired reflectance

+
+ 		+ -16 +
+ m + + float + +
+

Polynomial order for the PML (imaginary part increases as distance ** m)

+
+ 		+ 4 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + + + + + +
TypeDescription
+ s_function_t + +
+

An s_function, which takes an ndarray (distances) and returns an ndarray (complex part

+
+
+ s_function_t + +
+

of the cell width; needs to be divided by sqrt(epilon_effective) * real(omega))

+
+
+ s_function_t + +
+

before use.

+
+
+ + +
+ +
+ +
+ + +

+ uniform_grid_scpml + + +

+
uniform_grid_scpml(
+    shape: Sequence[int],
+    thicknesses: Sequence[int],
+    omega: float,
+    epsilon_effective: float = 1.0,
+    s_function: s_function_t | None = None,
+) -> list[list[NDArray[numpy.float64]]]
+
+ +
+ +

Create dx arrays for a uniform grid with a cell width of 1 and a pml.

+

If you want something more fine-grained, check out stretch_with_scpml(...).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ shape + + Sequence[int] + +
+

Shape of the grid, including the PMLs (which are 2*thicknesses thick)

+
+ 		+ required +
+ thicknesses + + Sequence[int] + +
+

[th_x, th_y, th_z] + Thickness of the PML in each direction. + Both polarities are added. + Each th_ of pml is applied twice, once on each edge of the grid along the given axis. + th_* may be zero, in which case no pml is added.

+
+ 		+ required +
+ omega + + float + +
+

Angular frequency for the simulation

+
+ 		+ required +
+ epsilon_effective + + float + +
+

Effective epsilon of the PML. Match this to the material + at the edge of your grid. + Default 1.

+
+ 		+ 1.0 +
+ s_function + + s_function_t | None + +
+

created by prepare_s_function(...), allowing customization of pml parameters. + Default uses prepare_s_function() with no parameters.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ list[list[NDArray[float64]]] + +
+

Complex cell widths (dx_lists_mut) as discussed in meanas.fdmath.types.

+
+
+ + +
+ +
+ +
+ + +

+ stretch_with_scpml + + +

+
stretch_with_scpml(
+    dxes: list[list[NDArray[float64]]],
+    axis: int,
+    polarity: int,
+    omega: float,
+    epsilon_effective: float = 1.0,
+    thickness: int = 10,
+    s_function: s_function_t | None = None,
+) -> list[list[NDArray[numpy.float64]]]
+
+ +
+ +

Stretch dxes to contain a stretched-coordinate PML (SCPML) in one direction along one axis.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dxes + + list[list[NDArray[float64]]] + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ axis + + int + +
+

axis to stretch (0=x, 1=y, 2=z)

+
+ 		+ required +
+ polarity + + int + +
+

direction to stretch (-1 for -ve, +1 for +ve)

+
+ 		+ required +
+ omega + + float + +
+

Angular frequency for the simulation

+
+ 		+ required +
+ epsilon_effective + + float + +
+

Effective epsilon of the PML. Match this to the material at the + edge of your grid. Default 1.

+
+ 		+ 1.0 +
+ thickness + + int + +
+

number of cells to use for pml (default 10)

+
+ 		+ 10 +
+ s_function + + s_function_t | None + +
+

Created by prepare_s_function(...), allowing customization + of pml parameters. Default uses prepare_s_function() with no parameters.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ list[list[NDArray[float64]]] + +
+

Complex cell widths (dx_lists_mut) as discussed in meanas.fdmath.types.

+
+
+ list[list[NDArray[float64]]] + +
+

Multiple calls to this function may be necessary if multiple absorpbing boundaries are needed.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdfd.farfield + + +

+ +
+ +

Functions for performing near-to-farfield transformation (and the reverse).

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ near_to_farfield + + +

+
near_to_farfield(
+    E_near: cfdfield_t,
+    H_near: cfdfield_t,
+    dx: float,
+    dy: float,
+    padded_size: list[int] | int | None = None,
+) -> dict[str, Any]
+
+ +
+ +

Compute the farfield, i.e. the distribution of the fields after propagation + through several wavelengths of uniform medium.

+

The input fields should be complex phasors.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ E_near + + cfdfield_t + +
+

List of 2 ndarrays containing the 2D phasor field slices for the transverse + E fields (e.g. [Ex, Ey] for calculating the farfield toward the z-direction).

+
+ 		+ required +
+ H_near + + cfdfield_t + +
+

List of 2 ndarrays containing the 2D phasor field slices for the transverse + H fields (e.g. [Hx, hy] for calculating the farfield towrad the z-direction).

+
+ 		+ required +
+ dx + + float + +
+

Cell size along x-dimension, in units of wavelength.

+
+ 		+ required +
+ dy + + float + +
+

Cell size along y-dimension, in units of wavelength.

+
+ 		+ required +
+ padded_size + + list[int] | int | None + +
+

Shape of the output. A single integer n will be expanded to (n, n). + Powers of 2 are most efficient for FFT computation. + Default is the smallest power of 2 larger than the input, for each axis.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
TypeDescription
+ dict[str, Any] + +
+

Dict with keys

+
+
+ dict[str, Any] + +
+
    +
  • E_far: Normalized E-field farfield; multiply by + (i k exp(-i k r) / (4 pi r)) to get the actual field value.
    • +
+
+
+ dict[str, Any] + +
+
    +
  • H_far: Normalized H-field farfield; multiply by + (i k exp(-i k r) / (4 pi r)) to get the actual field value.
    • +
+
+
+ dict[str, Any] + +
+
    +
  • kx, ky: Wavevector values corresponding to the x- and y- axes in E_far and H_far, + normalized to wavelength (dimensionless).
    • +
+
+
+ dict[str, Any] + +
+
    +
  • dkx, dky: step size for kx and ky, normalized to wavelength.
    • +
+
+
+ dict[str, Any] + +
+
    +
  • theta: arctan2(ky, kx) corresponding to each (kx, ky). + This is the angle in the x-y plane, counterclockwise from above, starting from +x.
    • +
+
+
+ dict[str, Any] + +
+
    +
  • phi: arccos(kz / k) corresponding to each (kx, ky). + This is the angle away from +z.
    • +
+
+
+ + +
+ +
+ +
+ + +

+ far_to_nearfield + + +

+
far_to_nearfield(
+    E_far: cfdfield_t,
+    H_far: cfdfield_t,
+    dkx: float,
+    dky: float,
+    padded_size: list[int] | int | None = None,
+) -> dict[str, Any]
+
+ +
+ +

Compute the farfield, i.e. the distribution of the fields after propagation + through several wavelengths of uniform medium.

+

The input fields should be complex phasors.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ E_far + + cfdfield_t + +
+

List of 2 ndarrays containing the 2D phasor field slices for the transverse + E fields (e.g. [Ex, Ey] for calculating the nearfield toward the z-direction). + Fields should be normalized so that + E_far = E_far_actual / (i k exp(-i k r) / (4 pi r))

+
+ 		+ required +
+ H_far + + cfdfield_t + +
+

List of 2 ndarrays containing the 2D phasor field slices for the transverse + H fields (e.g. [Hx, hy] for calculating the nearfield toward the z-direction). + Fields should be normalized so that + H_far = H_far_actual / (i k exp(-i k r) / (4 pi r))

+
+ 		+ required +
+ dkx + + float + +
+

kx discretization, in units of wavelength.

+
+ 		+ required +
+ dky + + float + +
+

ky discretization, in units of wavelength.

+
+ 		+ required +
+ padded_size + + list[int] | int | None + +
+

Shape of the output. A single integer n will be expanded to (n, n). + Powers of 2 are most efficient for FFT computation. + Default is the smallest power of 2 larger than the input, for each axis.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + + + + + + + + + +
TypeDescription
+ dict[str, Any] + +
+

Dict with keys

+
+
+ dict[str, Any] + +
+
    +
  • E: E-field nearfield
    • +
+
+
+ dict[str, Any] + +
+
    +
  • H: H-field nearfield
    • +
+
+
+ dict[str, Any] + +
+
    +
  • dx, dy: spatial discretization, normalized to wavelength (dimensionless)
    • +
+
+
+ + +
+ +
+ + + +
+ +
+ +
+ + + + + + + + + + + + + +
+
+ + + +
+ + + +
+ + + +
+
+
+
+ + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/api/fdmath/index.html b/api/fdmath/index.html new file mode 100644 index 0000000..61228f3 --- /dev/null +++ b/api/fdmath/index.html @@ -0,0 +1,4603 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + fdmath - meanas + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + Skip to content + + +
+
+ +
+ + + + + + +
+ + +
+ +
+ + + + + + +
+
+ + + +
+
+
+ + + + + +
+
+
+ + + +
+
+
+ + + +
+
+
+ + + +
+ +
+ + + + + +

fdmath

+ + +
+ + + +

+ meanas.fdmath + + +

+ +
+ +

Basic discrete calculus for finite difference (fd) simulations.

+

Fields, Functions, and Operators

+

Discrete fields are stored in one of two forms:

+
    +
  • The fdfield_t form is a multidimensional numpy.NDArray
      +
    • For a scalar field, this is just U[m, n, p], where m, n, and p are + discrete indices referring to positions on the x, y, and z axes respectively.
      • +
    • For a vector field, the first index specifies which vector component is accessed: + E[:, m, n, p] = [Ex[m, n, p], Ey[m, n, p], Ez[m, n, p]].
      • +
    +
    • +
  • The vfdfield_t form is simply a vectorzied (i.e. 1D) version of the fdfield_t, + as obtained by meanas.fdmath.vectorization.vec (effectively just numpy.ravel)
    • +
+ + +
+ Operators which act on fields also come in two forms +
    +
  • Python functions, created by the functions in meanas.fdmath.functional. + The generated functions act on fields in the fdfield_t form.
    • +
  • Linear operators, usually 2D sparse matrices using scipy.sparse, created + by meanas.fdmath.operators. These operators act on vectorized fields in the + vfdfield_t form.
    • +
+

The operations performed should be equivalent: functional.op(*args)(E) should be +equivalent to unvec(operators.op(*args) @ vec(E), E.shape[1:]).

+

Generally speaking the field_t form is easier to work with, but can be harder or less +efficient to compose (e.g. it is easy to generate a single matrix by multiplying a +series of other matrices).

+

Discrete calculus

+

This documentation and approach is roughly based on W.C. Chew's excellent +"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770), +which covers a superset of this material with similar notation and more detail.

+

Scalar derivatives and cell shifts

+

Define the discrete forward derivative as + $$ [\tilde{\partial}x f] - f_m) $$ + where }{2}} = \frac{1}{\Delta_{x, m}} (f_{m + 1\(f\) is a function defined at discrete locations on the x-axis (labeled using \(m\)). + The value at \(m\) occupies a length \(\Delta_{x, m}\) along the x-axis. Note that \(m\) + is an index along the x-axis, not necessarily an x-coordinate, since each length + \(\Delta_{x, m}, \Delta_{x, m+1}, ...\) is independently chosen.

+

If we treat f as a 1D array of values, with the i-th value f[i] taking up a length dx[i] +along the x-axis, the forward derivative is

+
deriv_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
+
+

Likewise, discrete reverse derivative is + $$ [\hat{\partial}x f ]) $$ + or}{2}} = \frac{1}{\Delta_{x, m}} (f_{m} - f_{m - 1

+
deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
+
+

The derivatives' values are shifted by a half-cell relative to the original function, and +will have different cell widths if all the dx[i] ( \(\Delta_{x, m}\) ) are not +identical:

+
[figure: derivatives and cell sizes]
+    dx0   dx1      dx2      dx3      cell sizes for function
+   ----- ----- ----------- -----
+   ______________________________
+        |     |           |     |
+     f0 |  f1 |     f2    |  f3 |    function
+   _____|_____|___________|_____|
+     |     |        |        |
+     | Df0 |   Df1  |   Df2  | Df3   forward derivative (periodic boundary)
+   __|_____|________|________|___
+
+ dx'3] dx'0   dx'1     dx'2  [dx'3   cell sizes for forward derivative
+   -- ----- -------- -------- ---
+ dx'0] dx'1   dx'2     dx'3  [dx'0   cell sizes for reverse derivative
+   ______________________________
+     |     |        |        |
+     | df1 |  df2   |   df3  | df0   reverse derivative (periodic boundary)
+   __|_____|________|________|___
+
+Periodic boundaries are used here and elsewhere unless otherwise noted.
+
+

In the above figure, + f0 = \(f_0\), f1 = \(f_1\) + Df0 = \([\tilde{\partial}f]_{0 + \frac{1}{2}}\) + Df1 = \([\tilde{\partial}f]_{1 + \frac{1}{2}}\) + df0 = \([\hat{\partial}f]_{0 - \frac{1}{2}}\) + etc.

+

The fractional subscript \(m + \frac{1}{2}\) is used to indicate values defined + at shifted locations relative to the original \(m\), with corresponding lengths + $$ \Delta_{x, m + \frac{1}{2}} = \frac{1}{2} * (\Delta_{x, m} + \Delta_{x, m + 1}) $$

+

Just as \(m\) is not itself an x-coordinate, neither is \(m + \frac{1}{2}\); +carefully note the positions of the various cells in the above figure vs their labels. +If the positions labeled with \(m\) are considered the "base" or "original" grid, +the positions labeled with \(m + \frac{1}{2}\) are said to lie on a "dual" or +"derived" grid.

+

For the remainder of the Discrete calculus section, all figures will show +constant-length cells in order to focus on the vector derivatives themselves. +See the Grid description section below for additional information on this topic +and generalization to three dimensions.

+

Gradients and fore-vectors

+

Expanding to three dimensions, we can define two gradients + $$ [\tilde{\nabla} f]{m,n,p} = \vec{x} [\tilde{\partial}_x f] + + \vec{y} [\tilde{\partial}}{2},n,py f] + + \vec{z} [\tilde{\partial}}{2},pz f] $$ + $$ [\hat{\nabla} f]}{2}{m,n,p} = \vec{x} [\hat{\partial}_x f] + + \vec{y} [\hat{\partial}}{2},n,py f] + + \vec{z} [\hat{\partial}}{2},pz f] $$}{2}

+

or

+
[code: gradients]
+grad_forward(f)[i,j,k] = [Dx_forward(f)[i, j, k],
+                          Dy_forward(f)[i, j, k],
+                          Dz_forward(f)[i, j, k]]
+                       = [(f[i + 1, j, k] - f[i, j, k]) / dx[i],
+                          (f[i, j + 1, k] - f[i, j, k]) / dy[i],
+                          (f[i, j, k + 1] - f[i, j, k]) / dz[i]]
+
+grad_back(f)[i,j,k] = [Dx_back(f)[i, j, k],
+                       Dy_back(f)[i, j, k],
+                       Dz_back(f)[i, j, k]]
+                    = [(f[i, j, k] - f[i - 1, j, k]) / dx[i],
+                       (f[i, j, k] - f[i, j - 1, k]) / dy[i],
+                       (f[i, j, k] - f[i, j, k - 1]) / dz[i]]
+
+

The three derivatives in the gradient cause shifts in different +directions, so the x/y/z components of the resulting "vector" are defined +at different points: the x-component is shifted in the x-direction, +y in y, and z in z.

+

We call the resulting object a "fore-vector" or "back-vector", depending +on the direction of the shift. We write it as + $$ \tilde{g}{m,n,p} = \vec{x} g^x + + \vec{y} g^y_{m,n + \frac{1}{2},p} + + \vec{z} g^z_{m,n,p + \frac{1}{2}} $$ + $$ \hat{g}}{2},n,p{m,n,p} = \vec{x} g^x + + \vec{y} g^y_{m,n - \frac{1}{2},p} + + \vec{z} g^z_{m,n,p - \frac{1}{2}} $$}{2},n,p

+
[figure: gradient / fore-vector]
+   (m, n+1, p+1) ______________ (m+1, n+1, p+1)
+                /:            /|
+               / :           / |
+              /  :          /  |
+  (m, n, p+1)/_____________/   |     The forward derivatives are defined
+             |   :         |   |     at the Dx, Dy, Dz points,
+             |   :.........|...|     but the forward-gradient fore-vector
+ z y        Dz  /          |  /      is the set of all three
+ |/_x        | Dy          | /       and is said to be "located" at (m,n,p)
+             |/            |/
+    (m, n, p)|_____Dx______| (m+1, n, p)
+
+

Divergences

+

There are also two divergences,

+

$$ d_{n,m,p} = [\tilde{\nabla} \cdot \hat{g}]{n,m,p} + = [\tilde{\partial}_x g^x] + + [\tilde{\partial}y g^y] + + [\tilde{\partial}z g^z] $$

+

$$ d_{n,m,p} = [\hat{\nabla} \cdot \tilde{g}]{n,m,p} + = [\hat{\partial}_x g^x] + + [\hat{\partial}y g^y] + + [\hat{\partial}z g^z] $$

+

or

+
[code: divergences]
+div_forward(g)[i,j,k] = Dx_forward(gx)[i, j, k] +
+                        Dy_forward(gy)[i, j, k] +
+                        Dz_forward(gz)[i, j, k]
+                      = (gx[i + 1, j, k] - gx[i, j, k]) / dx[i] +
+                        (gy[i, j + 1, k] - gy[i, j, k]) / dy[i] +
+                        (gz[i, j, k + 1] - gz[i, j, k]) / dz[i]
+
+div_back(g)[i,j,k] = Dx_back(gx)[i, j, k] +
+                     Dy_back(gy)[i, j, k] +
+                     Dz_back(gz)[i, j, k]
+                   = (gx[i, j, k] - gx[i - 1, j, k]) / dx[i] +
+                     (gy[i, j, k] - gy[i, j - 1, k]) / dy[i] +
+                     (gz[i, j, k] - gz[i, j, k - 1]) / dz[i]
+
+

where g = [gx, gy, gz] is a fore- or back-vector field.

+

Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value +is defined at the back-vector's (fore-vector's) location \((m,n,p)\) and not at the locations of its components +\((m \pm \frac{1}{2},n,p)\) etc.

+
[figure: divergence]
+                                ^^
+     (m-1/2, n+1/2, p+1/2) _____||_______ (m+1/2, n+1/2, p+1/2)
+                          /:    ||  ,,  /|
+                         / :    || //  / |      The divergence at (m, n, p) (the center
+                        /  :      //  /  |      of this cube) of a fore-vector field
+  (m-1/2, n-1/2, p+1/2)/_____________/   |      is the sum of the outward-pointing
+                       |   :         |   |      fore-vector components, which are
+     z y            <==|== :.........|.====>    located at the face centers.
+     |/_x              |  /          |  /
+                       | /    //     | /       Note that in a nonuniform grid, each
+                       |/    // ||   |/        dimension is normalized by the cell width.
+  (m-1/2, n-1/2, p-1/2)|____//_______| (m+1/2, n-1/2, p-1/2)
+                           ''   ||
+                                VV
+
+

Curls

+

The two curls are then

+

$$ \begin{aligned} + \hat{h}{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &= \ + [\tilde{\nabla} \times \tilde{g}] &= + \vec{x} (\tilde{\partial}}{2}, n + \frac{1}{2}, p + \frac{1}{2}y g^z}{2}} - \tilde{\partialz g^y) \ + &+ \vec{y} (\tilde{\partial}}{2},pz g^x}{2},n,p} - \tilde{\partialx g^z) \ + &+ \vec{z} (\tilde{\partial}}{2}x g^y}{2},p} - \tilde{\partialy g^z) + \end{aligned} $$}{2},n,p

+

and

+

$$ \tilde{h}{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} = + [\hat{\nabla} \times \hat{g}] $$}{2}, n - \frac{1}{2}, p - \frac{1}{2}

+

where \(\hat{g}\) and \(\tilde{g}\) are located at \((m,n,p)\) + with components at \((m \pm \frac{1}{2},n,p)\) etc., + while \(\hat{h}\) and \(\tilde{h}\) are located at \((m \pm \frac{1}{2}, n \pm \frac{1}{2}, p \pm \frac{1}{2})\) + with components at \((m, n \pm \frac{1}{2}, p \pm \frac{1}{2})\) etc.

+
[code: curls]
+curl_forward(g)[i,j,k] = [Dy_forward(gz)[i, j, k] - Dz_forward(gy)[i, j, k],
+                          Dz_forward(gx)[i, j, k] - Dx_forward(gz)[i, j, k],
+                          Dx_forward(gy)[i, j, k] - Dy_forward(gx)[i, j, k]]
+
+curl_back(g)[i,j,k] = [Dy_back(gz)[i, j, k] - Dz_back(gy)[i, j, k],
+                       Dz_back(gx)[i, j, k] - Dx_back(gz)[i, j, k],
+                       Dx_back(gy)[i, j, k] - Dy_back(gx)[i, j, k]]
+
+

For example, consider the forward curl, at (m, n, p), of a back-vector field g, defined + on a grid containing (m + 1/2, n + 1/2, p + 1/2). + The curl will be a fore-vector, so its z-component will be defined at (m, n, p + 1/2). + Take the nearest x- and y-components of g in the xy plane where the curl's z-component + is located; these are

+
[curl components]
+(m,       n + 1/2, p + 1/2) : x-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
+(m + 1,   n + 1/2, p + 1/2) : x-component of back-vector at (m + 3/2, n + 1/2, p + 1/2)
+(m + 1/2, n      , p + 1/2) : y-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
+(m + 1/2, n + 1  , p + 1/2) : y-component of back-vector at (m + 1/2, n + 3/2, p + 1/2)
+
+

These four xy-components can be used to form a loop around the curl's z-component; its magnitude and sign + is set by their loop-oriented sum (i.e. two have their signs flipped to complete the loop).

+
[figure: z-component of curl]
+                          :             |
+    z y                   :    ^^       |
+    |/_x                  :....||.<.....|  (m+1, n+1, p+1/2)
+                          /    ||      /
+                       | v     ||   | ^
+                       |/           |/
+         (m, n, p+1/2) |_____>______|  (m+1, n, p+1/2)
+
+

Maxwell's Equations

+

If we discretize both space (m,n,p) and time (l), Maxwell's equations become

+

$$ \begin{aligned} + \tilde{\nabla} \times \tilde{E}{l,\vec{r}} &= -\tilde{\partial}_t \hat{B} + - \hat{M}}{2}, \vec{r} + \frac{1}{2}{l, \vec{r} + \frac{1}{2}} \ + \hat{\nabla} \times \hat{H}}{2},\vec{r} + \frac{1}{2}} &= \hat{\partialt \tilde{D} + + \tilde{J}}{l-\frac{1}{2},\vec{r}} \ + \tilde{\nabla} \cdot \hat{B} &= 0 \ + \hat{\nabla} \cdot \tilde{D}}{2}, \vec{r} + \frac{1}{2}{l,\vec{r}} &= \rho + \end{aligned} $$}

+

with

+

$$ \begin{aligned} + \hat{B}{\vec{r}} &= \mu} + \frac{1}{2}} \cdot \hat{H{\vec{r} + \frac{1}{2}} \ + \tilde{D} + \end{aligned} $$}} &= \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}

+

where the spatial subscripts are abbreviated as \(\vec{r} = (m, n, p)\) and +\(\vec{r} + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2})\), +\(\tilde{E}\) and \(\hat{H}\) are the electric and magnetic fields, +\(\tilde{J}\) and \(\hat{M}\) are the electric and magnetic current distributions, +and \(\epsilon\) and \(\mu\) are the dielectric permittivity and magnetic permeability.

+

The above is Yee's algorithm, written in a form analogous to Maxwell's equations. +The time derivatives can be expanded to form the update equations:

+
[code: Maxwell's equations updates]
+H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) /      mu[i, j, k]
+E[i, j, k] += dt * (curl_back(   H)[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
+
+

Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting +in distinct locations for all six E- and H-field components:

+
[figure: Field components]
+
+        (m - 1/2,=> ____________Hx__________[H] <= r + 1/2 = (m + 1/2,
+         n + 1/2,  /:           /:          /|                n + 1/2,
+   z y   p + 1/2) / :          / :         / |                p + 1/2)
+   |/_x          /  :         /  :        /  |
+                /   :       Ez__________Hy   |      Locations of the E- and
+               /    :        :   :      /|   |      H-field components for the
+ (m - 1/2,    /     :        :  Ey...../.|..Hz      [E] fore-vector at r = (m,n,p)
+  n - 1/2, =>/________________________/  |  /|      (the large cube's center)
+  p + 1/2)   |      :        : /      |  | / |      and [H] back-vector at r + 1/2
+             |      :        :/       |  |/  |      (the top right corner)
+             |      :       [E].......|.Ex   |
+             |      :.................|......| <= (m + 1/2, n + 1/2, p + 1/2)
+             |     /                  |     /
+             |    /                   |    /
+             |   /                    |   /         This is the Yee discretization
+             |  /                     |  /          scheme ("Yee cell").
+r - 1/2 =    | /                      | /
+ (m - 1/2,   |/                       |/
+  n - 1/2,=> |________________________| <= (m + 1/2, n - 1/2, p - 1/2)
+  p - 1/2)
+
+

Each component forms its own grid, offset from the others:

+
[figure: E-fields for adjacent cells]
+
+              H1__________Hx0_________H0
+  z y        /:                       /|
+  |/_x      / :                      / |    This figure shows H back-vector locations
+           /  :                     /  |    H0, H1, etc. and their associated components
+         Hy1  :                   Hy0  |    H0 = (Hx0, Hy0, Hz0) etc.
+         /    :                   /    |
+        /    Hz1                 /     Hz0
+       H2___________Hx3_________H3     |    The equivalent drawing for E would have
+       |      :                 |      |    fore-vectors located at the cube's
+       |      :                 |      |    center (and the centers of adjacent cubes),
+       |      :                 |      |    with components on the cube's faces.
+       |      H5..........Hx4...|......H4
+       |     /                  |     /
+      Hz2   /                  Hz2   /
+       |   /                    |   /
+       | Hy6                    | Hy4
+       | /                      | /
+       |/                       |/
+       H6__________Hx7__________H7
+
+

The divergence equations can be derived by taking the divergence of the curl equations +and combining them with charge continuity, + $$ \hat{\nabla} \cdot \tilde{J} + \hat{\partial}_t \rho = 0 $$ + implying that the discrete Maxwell's equations do not produce spurious charges.

+

Wave equation

+

Taking the backward curl of the \(\tilde{\nabla} \times \tilde{E}\) equation and +replacing the resulting \(\hat{\nabla} \times \hat{H}\) term using its respective equation, +and setting \(\hat{M}\) to zero, we can form the discrete wave equation:

+
\[ + \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= + -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} + - \hat{M}_{l-1, \vec{r} + \frac{1}{2}} \\ + \mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= + -\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &= + \hat{\nabla} \times (-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}) \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &= + -\tilde{\partial}_t \hat{\nabla} \times \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &= + -\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} + \hat{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) + + \tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \cdot \tilde{E}_{l, \vec{r}} + &= \tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}} + \end{aligned} +\]
+

Frequency domain

+

We can substitute in a time-harmonic fields

+
\[ + \begin{aligned} + \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\ + \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} + \end{aligned} +\]
+

resulting in

+
\[ + \begin{aligned} + \tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\ + \hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\ + \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t + \end{aligned} +\]
+

This gives the frequency-domain wave equation,

+
\[ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}}) + -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\ +\]
+

Plane waves and Dispersion relation

+

With uniform material distribution and no sources

+
\[ + \begin{aligned} + \mu_{\vec{r} + \frac{1}{2}} &= \mu \\ + \epsilon_{\vec{r}} &= \epsilon \\ + \tilde{J}_{\vec{r}} &= 0 \\ + \end{aligned} +\]
+

the frequency domain wave equation simplifies to

+
\[ \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} - \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 \]
+

Since \(\hat{\nabla} \cdot \tilde{E}_{\vec{r}} = 0\), we can simplify

+
\[ + \begin{aligned} + \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} + &= \tilde{\nabla}(\hat{\nabla} \cdot \tilde{E}_{\vec{r}}) - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\ + &= - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\ + &= - \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \end{aligned} +\]
+

and we get

+
\[ \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 \]
+

We can convert this to three scalar-wave equations of the form

+
\[ (\tilde{\nabla}^2 + K^2) \phi_{\vec{r}} = 0 \]
+

with \(K^2 = \Omega^2 \mu \epsilon\). Now we let

+
\[ \phi_{\vec{r}} = A e^{\imath (k_x m \Delta_x + k_y n \Delta_y + k_z p \Delta_z)} \]
+

resulting in

+
\[ + \begin{aligned} + \tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\ + \hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\ + K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\ + \end{aligned} +\]
+

with similar expressions for the y and z dimnsions (and \(K_y, K_z\)).

+

This implies

+
\[ + \tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\ + K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2 +\]
+

where \(c = \sqrt{\mu \epsilon}\).

+

Assuming real \((k_x, k_y, k_z), \omega\) will be real only if

+
\[ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) \]
+

If \(\Delta_x = \Delta_y = \Delta_z\), this simplifies to \(c \Delta_t < \Delta_x / \sqrt{3}\). +This last form can be interpreted as enforcing causality; the distance that light +travels in one timestep (i.e., \(c \Delta_t\)) must be less than the diagonal +of the smallest cell ( \(\Delta_x / \sqrt{3}\) when on a uniform cubic grid).

+

Grid description

+

As described in the section on scalar discrete derivatives above, cell widths +(dx[i], dy[j], dz[k]) along each axis can be arbitrary and independently +defined. Moreover, all field components are actually defined at "derived" or "dual" +positions, in-between the "base" grid points on one or more axes.

+

To get a better sense of how this works, let's start by drawing a grid with uniform +dy and dz and nonuniform dx. We will only draw one cell in the y and z dimensions +to make the illustration simpler; we need at least two cells in the x dimension to +demonstrate how nonuniform dx affects the various components.

+

Place the E fore-vectors at integer indices \(r = (m, n, p)\) and the H back-vectors +at fractional indices \(r + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, +p + \frac{1}{2})\). Remember that these are indices and not coordinates; they can +correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.

+

Draw lines to denote the planes on which the H components and back-vectors are defined. +For simplicity, don't draw the equivalent planes for the E components and fore-vectors, +except as necessary to show their locations -- it's easiest to just connect them to their +associated H-equivalents.

+

The result looks something like this:

+
[figure: Component centers]
+                                                             p=
+          [H]__________Hx___________[H]_____Hx______[H]   __ +1/2
+  z y     /:           /:           /:      /:      /|     |      |
+  |/_x   / :          / :          / :     / :     / |     |      |
+        /  :         /  :         /  :    /  :    /  |     |      |
+      Hy   :       Ez...........Hy   :  Ez......Hy   |     |      |
+      /:   :        :   :       /:   :   :   :  /|   |     |      |
+     / :  Hz        :  Ey....../.:..Hz   :  Ey./.|..Hz    __ 0    | dz[0]
+    /  :  /:        :  /      /  :  /:   :  / /  |  /|     |      |
+   /_________________________/_______________/   | / |     |      |
+   |   :/  :        :/       |   :/  :   :/  |   |/  |     |      |
+   |  Ex   :       [E].......|..Ex   :  [E]..|..Ex   |     |      |
+   |       :                 |       :       |       |     |      |
+   |      [H]..........Hx....|......[H].....H|x.....[H]   __ --------- (n=+1/2, p=-1/2)
+   |      /                  |      /        |      /     /       /
+  Hz     /                  Hz     /        Hz     /     /       /
+   |    /                    |    /          |    /     /       /
+   |  Hy                     |  Hy           |  Hy    __ 0     / dy[0]
+   |  /                      |  /            |  /     /       /
+   | /                       | /             | /     /       /
+   |/                        |/              |/     /       /
+  [H]__________Hx___________[H]_____Hx______[H]   __ -1/2  /
+                                                       =n
+   |------------|------------|-------|-------|
+ -1/2           0          +1/2     +1     +3/2 = m
+
+    ------------------------- ----------------
+              dx[0]                  dx[1]
+
+  Part of a nonuniform "base grid", with labels specifying
+  positions of the various field components. [E] fore-vectors
+  are at the cell centers, and [H] back-vectors are at the
+  vertices. H components along the near (-y) top (+z) edge
+  have been omitted to make the insides of the cubes easier
+  to visualize.
+
+

The above figure shows where all the components are located; however, it is also useful to show +what volumes those components correspond to. Consider the Ex component at m = +1/2: it is +shifted in the x-direction by a half-cell from the E fore-vector at m = 0 (labeled [E] +in the figure). It corresponds to a volume between m = 0 and m = +1 (the other +dimensions are not shifted, i.e. they are still bounded by n, p = +-1/2). (See figure +below). Since m is an index and not an x-coordinate, the Ex component is not necessarily +at the center of the volume it represents, and the x-length of its volume is the derived +quantity dx'[0] = (dx[0] + dx[1]) / 2 rather than the base dx. +(See also Scalar derivatives and cell shifts).

+
[figure: Ex volumes]
+                                                             p=
+           <_________________________________________>   __ +1/2
+  z y     <<           /:           /       /:      >>    |      |
+  |/_x   < <          / :          /       / :     > >    |      |
+        <  <         /  :         /       /  :    >  >    |      |
+       <   <        /   :        /       /   :   >   >    |      |
+      <:   <       /    :        :      /    :  >:   >    |      |
+     < :   <      /     :        :     /     : > :   >   __ 0    | dz[0]
+    <  :   <     /      :        :    /      :>  :   >    |      |
+   <____________/____________________/_______>   :   >    |      |
+   <   :   <    |       :        :   |       >   :   >    |      |
+   <  Ex   <    |       :       Ex   |       >  Ex   >    |      |
+   <   :   <    |       :        :   |       >   :   >    |      |
+   <   :   <....|.......:........:...|.......>...:...>   __ --------- (n=+1/2, p=-1/2)
+   <   :  <     |      /         :  /|      />   :  >    /       /
+   <   : <      |     /          : / |     / >   : >    /       /
+   <   :<       |    /           :/  |    /  >   :>    /       /
+   <   <        |   /            :   |   /   >   >    _ 0     / dy[0]
+   <  <         |  /                 |  /    >  >    /       /
+   < <          | /                  | /     > >    /       /
+   <<           |/                   |/      >>    /       /
+   <____________|____________________|_______>   __ -1/2  /
+                                                     =n
+   |------------|------------|-------|-------|
+ -1/2           0          +1/2      +1    +3/2 = m
+
+   ~------------ -------------------- -------~
+     dx'[-1]          dx'[0]           dx'[1]
+
+ The Ex values are positioned on the x-faces of the base
+ grid. They represent the Ex field in volumes shifted by
+ a half-cell in the x-dimension, as shown here. Only the
+ center cell (with width dx'[0]) is fully shown; the
+ other two are truncated (shown using >< markers).
+
+ Note that the Ex positions are the in the same positions
+ as the previous figure; only the cell boundaries have moved.
+ Also note that the points at which Ex is defined are not
+ necessarily centered in the volumes they represent; non-
+ uniform cell sizes result in off-center volumes like the
+ center cell here.
+
+

The next figure shows the volumes corresponding to the Hy components, which +are shifted in two dimensions (x and z) compared to the base grid.

+
[figure: Hy volumes]
+                                                             p=
+  z y     mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm   __ +1/2  s
+  |/_x   <<           m:                    m:      >>    |       |
+        < <          m :                   m :     > >    |       | dz'[1]
+       <  <         m  :                  m  :    >  >    |       |
+     Hy........... m........Hy...........m......Hy   >    |       |
+     <    <       m    :                m    :  >    >    |       |
+    <     ______ m_____:_______________m_____:_>______   __ 0
+   <      <     m     /:              m     / >      >    |       |
+  mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm       >    |       |
+  <       <    |    /  :             |    /  >       >    |       | dz'[0]
+  <       <    |   /   :             |   /   >       >    |       |
+  <       <    |  /    :             |  /    >       >    |       |
+  <       wwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwwww   __       s
+  <      <     |/     w              |/     w>      >    /         /
+  _____________|_____________________|________     >    /         /
+  <    <       |    w                |    w  >    >    /         /
+  <  Hy........|...w........Hy.......|...w...>..Hy    _ 0       / dy[0]
+  < <          |  w                  |  w    >  >    /         /
+  <<           | w                   | w     > >    /         /
+  <            |w                    |w      >>    /         /
+  wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww   __ -1/2    /
+
+  |------------|------------|--------|-------|
+-1/2           0          +1/2      +1     +3/2 = m
+
+  ~------------ --------------------- -------~
+     dx'[-1]            dx'[0]         dx'[1]
+
+ The Hy values are positioned on the y-edges of the base
+ grid. Again here, the 'Hy' labels represent the same points
+ as in the basic grid figure above; the edges have shifted
+ by a half-cell along the x- and z-axes.
+
+ The grid lines _|:/ are edges of the area represented by
+ each Hy value, and the lines drawn using <m>.w represent
+ edges where a cell's faces extend beyond the drawn area
+ (i.e. where the drawing is truncated in the x- or z-
+ directions).
+
+

Datastructure: dx_lists_t

+

In this documentation, the E fore-vectors are placed on the base grid. An +equivalent formulation could place the H back-vectors on the base grid instead. +However, in the case of a non-uniform grid, the operation to get from the "base" +cell widths to "derived" ones is not its own inverse.

+

The base grid's cell sizes could be fully described by a list of three 1D arrays, +specifying the cell widths along all three axes:

+
[dx, dy, dz] = [[dx[0], dx[1], ...], [dy[0], ...], [dz[0], ...]]
+
+

Note that this is a list-of-arrays rather than a 2D array, as the simulation domain +may have a different number of cells along each axis.

+

Knowing the base grid's cell widths and the boundary conditions (periodic unless +otherwise noted) is enough information to calculate the cell widths dx', dy', +and dz' for the derived grids.

+

However, since most operations are trivially generalized to allow either E or H +to be defined on the base grid, they are written to take the a full set of base +and derived cell widths, distinguished by which field they apply to rather than +their "base" or "derived" status. This removes the need for each function to +generate the derived widths, and makes the "base" vs "derived" distinction +unnecessary in the code.

+

The resulting data structure containing all the cell widths takes the form of a +list-of-lists-of-arrays. The first list-of-arrays provides the cell widths for +the E-field fore-vectors, while the second list-of-arrays does the same for the +H-field back-vectors:

+
 [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
+  [[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
+
+

where dx_e[0] is the x-width of the m=0 cells, as used when calculating dE/dx, + and dy_h[0] is the y-width of the n=0 cells, as used when calculating dH/dy, etc.

+

Permittivity and Permeability

+

Since each vector component of E and H is defined in a different location and represents +a different volume, the value of the spatially-discrete epsilon and mu can also be +different for all three field components, even when representing a simple planar interface +between two isotropic materials.

+

As a result, epsilon and mu are taken to have the same dimensions as the field, and +composed of the three diagonal tensor components:

+
[equations: epsilon_and_mu]
+epsilon = [epsilon_xx, epsilon_yy, epsilon_zz]
+mu = [mu_xx, mu_yy, mu_zz]
+
+

or

+
\[ + \epsilon = \begin{bmatrix} \epsilon_{xx} & 0 & 0 \\ + 0 & \epsilon_{yy} & 0 \\ + 0 & 0 & \epsilon_{zz} \end{bmatrix} +$$ +$$ + \mu = \begin{bmatrix} \mu_{xx} & 0 & 0 \\ + 0 & \mu_{yy} & 0 \\ + 0 & 0 & \mu_{zz} \end{bmatrix} +\]
+

where the off-diagonal terms (e.g. epsilon_xy) are assumed to be zero.

+

High-accuracy volumetric integration of shapes on multiple grids can be performed +by the gridlock module.

+

The values of the vacuum permittivity and permability effectively become scaling +factors that appear in several locations (e.g. between the E and H fields). In +order to limit floating-point inaccuracy and simplify calculations, they are often +set to 1 and relative permittivities and permeabilities are used in their places; +the true values can be multiplied back in after the simulation is complete if non- +normalized results are needed.

+ + + + + + + + + + +
+ + + + + + + + + + + + +
+ +
+ +

Functional and sparse operators

+ + +
+ + + +

+ meanas.fdmath.functional + + +

+ +
+ +

Math functions for finite difference simulations

+

Basic discrete calculus etc.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ deriv_forward + + +

+
deriv_forward(
+    dx_e: Sequence[NDArray[floating | complexfloating]]
+    | None = None,
+) -> tuple[
+    fdfield_updater_t, fdfield_updater_t, fdfield_updater_t
+]
+
+ +
+ +

Utility operators for taking discretized derivatives (backward variant).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_e + + Sequence[NDArray[floating | complexfloating]] | None + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[fdfield_updater_t, fdfield_updater_t, fdfield_updater_t] + +
+

List of functions for taking forward derivatives along each axis.

+
+
+ + +
+ +
+ +
+ + +

+ deriv_back + + +

+
deriv_back(
+    dx_h: Sequence[NDArray[floating | complexfloating]]
+    | None = None,
+) -> tuple[
+    fdfield_updater_t, fdfield_updater_t, fdfield_updater_t
+]
+
+ +
+ +

Utility operators for taking discretized derivatives (forward variant).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_h + + Sequence[NDArray[floating | complexfloating]] | None + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[fdfield_updater_t, fdfield_updater_t, fdfield_updater_t] + +
+

List of functions for taking forward derivatives along each axis.

+
+
+ + +
+ +
+ +
+ + +

+ curl_forward + + +

+
curl_forward(
+    dx_e: Sequence[NDArray[floating | complexfloating]]
+    | None = None,
+) -> Callable[[TT], TT]
+
+ +
+ +

Curl operator for use with the E field.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_e + + Sequence[NDArray[floating | complexfloating]] | None + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ Callable[[TT], TT] + +
+

Function f for taking the discrete forward curl of a field,

+
+
+ Callable[[TT], TT] + +
+

f(E) -> curlE \(= \nabla_f \times E\)

+
+
+ + +
+ +
+ +
+ + +

+ curl_back + + +

+
curl_back(
+    dx_h: Sequence[NDArray[floating | complexfloating]]
+    | None = None,
+) -> Callable[[TT], TT]
+
+ +
+ +

Create a function which takes the backward curl of a field.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_h + + Sequence[NDArray[floating | complexfloating]] | None + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ Callable[[TT], TT] + +
+

Function f for taking the discrete backward curl of a field,

+
+
+ Callable[[TT], TT] + +
+

f(H) -> curlH \(= \nabla_b \times H\)

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdmath.operators + + +

+ +
+ +

Matrix operators for finite difference simulations

+

Basic discrete calculus etc.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ shift_circ + + +

+
shift_circ(
+    axis: int, shape: Sequence[int], shift_distance: int = 1
+) -> sparse.sparray
+
+ +
+ +

Utility operator for performing a circular shift along a specified axis by a + specified number of elements.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ axis + + int + +
+

Axis to shift along. x=0, y=1, z=2

+
+ 		+ required +
+ shape + + Sequence[int] + +
+

Shape of the grid being shifted

+
+ 		+ required +
+ shift_distance + + int + +
+

Number of cells to shift by. May be negative. Default 1.

+
+ 		+ 1 +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for performing the circular shift.

+
+
+ + +
+ +
+ +
+ + +

+ shift_with_mirror + + +

+
shift_with_mirror(
+    axis: int, shape: Sequence[int], shift_distance: int = 1
+) -> sparse.sparray
+
+ +
+ +

Utility operator for performing an n-element shift along a specified axis, with mirror +boundary conditions applied to the cells beyond the receding edge.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ axis + + int + +
+

Axis to shift along. x=0, y=1, z=2

+
+ 		+ required +
+ shape + + Sequence[int] + +
+

Shape of the grid being shifted

+
+ 		+ required +
+ shift_distance + + int + +
+

Number of cells to shift by. May be negative. Default 1.

+
+ 		+ 1 +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for performing the shift-with-mirror.

+
+
+ + +
+ +
+ +
+ + +

+ deriv_forward + + +

+
deriv_forward(
+    dx_e: Sequence[NDArray[floating | complexfloating]],
+) -> list[sparse.sparray]
+
+ +
+ +

Utility operators for taking discretized derivatives (forward variant).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_e + + Sequence[NDArray[floating | complexfloating]] + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ list[sparray] + +
+

List of operators for taking forward derivatives along each axis.

+
+
+ + +
+ +
+ +
+ + +

+ deriv_back + + +

+
deriv_back(
+    dx_h: Sequence[NDArray[floating | complexfloating]],
+) -> list[sparse.sparray]
+
+ +
+ +

Utility operators for taking discretized derivatives (backward variant).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_h + + Sequence[NDArray[floating | complexfloating]] + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ list[sparray] + +
+

List of operators for taking forward derivatives along each axis.

+
+
+ + +
+ +
+ +
+ + +

+ cross + + +

+
cross(B: Sequence[sparray]) -> sparse.sparray
+
+ +
+ +

Cross product operator

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ B + + Sequence[sparray] + +
+

List [Bx, By, Bz] of sparse matrices corresponding to the x, y, z +portions of the operator on the left side of the cross product.

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix corresponding to (B x), where x is the cross product.

+
+
+ + +
+ +
+ +
+ + +

+ vec_cross + + +

+
vec_cross(b: vfdfield_t) -> sparse.sparray
+
+ +
+ +

Vector cross product operator

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ b + + vfdfield_t + +
+

Vector on the left side of the cross product.

+
+ 		+ required +
+

Returns:

+
Sparse matrix corresponding to (b x), where x is the cross product.
+
+ + +
+ +
+ +
+ + +

+ avg_forward + + +

+
avg_forward(
+    axis: int, shape: Sequence[int]
+) -> sparse.sparray
+
+ +
+ +

Forward average operator (x4 = (x4 + x5) / 2)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ axis + + int + +
+

Axis to average along (x=0, y=1, z=2)

+
+ 		+ required +
+ shape + + Sequence[int] + +
+

Shape of the grid to average

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for forward average operation.

+
+
+ + +
+ +
+ +
+ + +

+ avg_back + + +

+
avg_back(axis: int, shape: Sequence[int]) -> sparse.sparray
+
+ +
+ +

Backward average operator (x4 = (x4 + x3) / 2)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ axis + + int + +
+

Axis to average along (x=0, y=1, z=2)

+
+ 		+ required +
+ shape + + Sequence[int] + +
+

Shape of the grid to average

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for backward average operation.

+
+
+ + +
+ +
+ +
+ + +

+ curl_forward + + +

+
curl_forward(
+    dx_e: Sequence[NDArray[floating | complexfloating]],
+) -> sparse.sparray
+
+ +
+ +

Curl operator for use with the E field.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_e + + Sequence[NDArray[floating | complexfloating]] + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for taking the discretized curl of the E-field

+
+
+ + +
+ +
+ +
+ + +

+ curl_back + + +

+
curl_back(
+    dx_h: Sequence[NDArray[floating | complexfloating]],
+) -> sparse.sparray
+
+ +
+ +

Curl operator for use with the H field.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dx_h + + Sequence[NDArray[floating | complexfloating]] + +
+

Lists of cell sizes for all axes + [[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...].

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix for taking the discretized curl of the H-field

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdmath.vectorization + + +

+ +
+ +

Functions for moving between a vector field (list of 3 ndarrays, [f_x, f_y, f_z]) +and a 1D array representation of that field [f_x0, f_x1, f_x2,... f_y0,... f_z0,...]. +Vectorized versions of the field use row-major (ie., C-style) ordering.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ vec + + +

+
+
vec(f: None) -> None
+
vec(f: fdfield_t) -> vfdfield_t
+
vec(f: cfdfield_t) -> vcfdfield_t
+
vec(f: fdfield2_t) -> vfdfield2_t
+
vec(f: cfdfield2_t) -> vcfdfield2_t
+
vec(f: fdslice_t) -> vfdslice_t
+
vec(f: cfdslice_t) -> vcfdslice_t
+
vec(f: ArrayLike) -> NDArray
+
+
vec(
+    f: fdfield_t
+    | cfdfield_t
+    | fdfield2_t
+    | cfdfield2_t
+    | fdslice_t
+    | cfdslice_t
+    | ArrayLike
+    | None,
+) -> (
+    vfdfield_t
+    | vcfdfield_t
+    | vfdfield2_t
+    | vcfdfield2_t
+    | vfdslice_t
+    | vcfdslice_t
+    | NDArray
+    | None
+)
+
+ +
+ +

Create a 1D ndarray from a vector field which spans a 1-3D region.

+

Returns None if called with f=None.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ f + + fdfield_t | cfdfield_t | fdfield2_t | cfdfield2_t | fdslice_t | cfdslice_t | ArrayLike | None + +
+

A vector field, e.g. [f_x, f_y, f_z] where each f_ component is a 1- to +3-D ndarray (f_* should all be the same size). Doesn't fail with f=None.

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ vfdfield_t | vcfdfield_t | vfdfield2_t | vcfdfield2_t | vfdslice_t | vcfdslice_t | NDArray | None + +
+

1D ndarray containing the linearized field (or None)

+
+
+ + +
+ +
+ +
+ + +

+ unvec + + +

+
+
unvec(
+    v: None, shape: Sequence[int], nvdim: int = 3
+) -> None
+
unvec(
+    v: vfdfield_t, shape: Sequence[int], nvdim: int = 3
+) -> fdfield_t
+
unvec(
+    v: vcfdfield_t, shape: Sequence[int], nvdim: int = 3
+) -> cfdfield_t
+
unvec(
+    v: vfdfield2_t, shape: Sequence[int], nvdim: int = 3
+) -> fdfield2_t
+
unvec(
+    v: vcfdfield2_t, shape: Sequence[int], nvdim: int = 3
+) -> cfdfield2_t
+
unvec(
+    v: vfdslice_t, shape: Sequence[int], nvdim: int = 3
+) -> fdslice_t
+
unvec(
+    v: vcfdslice_t, shape: Sequence[int], nvdim: int = 3
+) -> cfdslice_t
+
unvec(
+    v: ArrayLike, shape: Sequence[int], nvdim: int = 3
+) -> NDArray
+
+
unvec(
+    v: vfdfield_t
+    | vcfdfield_t
+    | vfdfield2_t
+    | vcfdfield2_t
+    | vfdslice_t
+    | vcfdslice_t
+    | ArrayLike
+    | None,
+    shape: Sequence[int],
+    nvdim: int = 3,
+) -> (
+    fdfield_t
+    | cfdfield_t
+    | fdfield2_t
+    | cfdfield2_t
+    | fdslice_t
+    | cfdslice_t
+    | NDArray
+    | None
+)
+
+ +
+ +

Perform the inverse of vec(): take a 1D ndarray and output an nvdim-component field + of form e.g. [f_x, f_y, f_z] (nvdim=3) where each of f_* is a len(shape)-dimensional + ndarray.

+

Returns None if called with v=None.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ v + + vfdfield_t | vcfdfield_t | vfdfield2_t | vcfdfield2_t | vfdslice_t | vcfdslice_t | ArrayLike | None + +
+

1D ndarray representing a vector field of shape shape (or None)

+
+ 		+ required +
+ shape + + Sequence[int] + +
+

shape of the vector field

+
+ 		+ required +
+ nvdim + + int + +
+

Number of components in each vector

+
+ 		+ 3 +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_t | cfdfield_t | fdfield2_t | cfdfield2_t | fdslice_t | cfdslice_t | NDArray | None + +
+

[f_x, f_y, f_z] where each f_ is a len(shape) dimensional ndarray (or None)

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdmath.types + + +

+ +
+ +

Types shared across multiple submodules

+ + + + + + + + + + +
+ + + + + + + +
+ + + +

+ dx_lists_t + + + + module-attribute + + +

+
dx_lists_t = Sequence[
+    Sequence[NDArray[floating | complexfloating]]
+]
+
+ +
+ +

'dxes' datastructure which contains grid cell width information in the following format:

+
[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
+ [[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
+
+

where dx_e[0] is the x-width of the x=0 cells, as used when calculating dE/dx, + and dy_h[0] is the y-width of the y=0 cells, as used when calculating dH/dy, etc.

+ +
+ +
+ +
+ + + +

+ dx_lists2_t + + + + module-attribute + + +

+
dx_lists2_t = Sequence[
+    Sequence[NDArray[floating | complexfloating]]
+]
+
+ +
+ +

2D 'dxes' datastructure which contains grid cell width information in the following format:

+
[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]],
+ [[dx_h[0], dx_h[1], ...], [dy_h[0], ...]]]
+
+

where dx_e[0] is the x-width of the x=0 cells, as used when calculating dE/dx, + and dy_h[0] is the y-width of the y=0 cells, as used when calculating dH/dy, etc.

+ +
+ +
+ +
+ + + +

+ dx_lists_mut + + + + module-attribute + + +

+
dx_lists_mut = MutableSequence[
+    MutableSequence[NDArray[floating | complexfloating]]
+]
+
+ +
+ +

Mutable version of dx_lists_t

+ +
+ +
+ +
+ + + +

+ dx_lists2_mut + + + + module-attribute + + +

+
dx_lists2_mut = MutableSequence[
+    MutableSequence[NDArray[floating | complexfloating]]
+]
+
+ +
+ +

Mutable version of dx_lists2_t

+ +
+ +
+ +
+ + + +

+ fdfield_updater_t + + + + module-attribute + + +

+
fdfield_updater_t = Callable[..., fdfield_t]
+
+ +
+ +

Convenience type for functions which take and return an fdfield_t

+ +
+ +
+ +
+ + + +

+ cfdfield_updater_t + + + + module-attribute + + +

+
cfdfield_updater_t = Callable[..., cfdfield_t]
+
+ +
+ +

Convenience type for functions which take and return an cfdfield_t

+ +
+ +
+ + +
+ + + +

+ fdfield + + +

+
fdfield = fdfield_t | NDArray[floating]
+
+ +
+ +

Vector field with shape (3, X, Y, Z) (e.g. [E_x, E_y, E_z])

+
+ +
+ +
+ + + +

+ vfdfield + + +

+
vfdfield = vfdfield_t | NDArray[floating]
+
+ +
+ +

Linearized vector field (single vector of length 3XY*Z)

+
+ +
+ +
+ + + +

+ cfdfield + + +

+
cfdfield = cfdfield_t | NDArray[complexfloating]
+
+ +
+ +

Complex vector field with shape (3, X, Y, Z) (e.g. [E_x, E_y, E_z])

+
+ +
+ +
+ + + +

+ vcfdfield + + +

+
vcfdfield = vcfdfield_t | NDArray[complexfloating]
+
+ +
+ +

Linearized complex vector field (single vector of length 3XY*Z)

+
+ +
+ +
+ + + +

+ fdslice + + +

+
fdslice = fdslice_t | NDArray[floating]
+
+ +
+ +

Vector field slice with shape (3, X, Y) (e.g. [E_x, E_y, E_z] at a single Z position)

+
+ +
+ +
+ + + +

+ vfdslice + + +

+
vfdslice = vfdslice_t | NDArray[floating]
+
+ +
+ +

Linearized vector field slice (single vector of length 3XY)

+
+ +
+ +
+ + + +

+ cfdslice + + +

+
cfdslice = cfdslice_t | NDArray[complexfloating]
+
+ +
+ +

Complex vector field slice with shape (3, X, Y) (e.g. [E_x, E_y, E_z] at a single Z position)

+
+ +
+ +
+ + + +

+ vcfdslice + + +

+
vcfdslice = vcfdslice_t | NDArray[complexfloating]
+
+ +
+ +

Linearized complex vector field slice (single vector of length 3XY)

+
+ +
+ +
+ + + +

+ fdfield2 + + +

+
fdfield2 = fdfield2_t | NDArray[floating]
+
+ +
+ +

2D Vector field with shape (2, X, Y) (e.g. [E_x, E_y])

+
+ +
+ +
+ + + +

+ vfdfield2 + + +

+
vfdfield2 = vfdfield2_t | NDArray[floating]
+
+ +
+ +

2D Linearized vector field (single vector of length 2XY)

+
+ +
+ +
+ + + +

+ cfdfield2 + + +

+
cfdfield2 = cfdfield2_t | NDArray[complexfloating]
+
+ +
+ +

2D Complex vector field with shape (2, X, Y) (e.g. [E_x, E_y])

+
+ +
+ +
+ + + +

+ vcfdfield2 + + +

+
vcfdfield2 = vcfdfield2_t | NDArray[complexfloating]
+
+ +
+ +

2D Linearized complex vector field (single vector of length 2XY)

+
+ +
+ + + + + +
+ +
+ +
+ + + + + + + + + + + + + +
+
+ + + +
+ + + +
+ + + +
+
+
+
+ + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/api/fdtd/index.html b/api/fdtd/index.html new file mode 100644 index 0000000..6511297 --- /dev/null +++ b/api/fdtd/index.html @@ -0,0 +1,3818 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + fdtd - meanas + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + Skip to content + + +
+
+ +
+ + + + + + +
+ + +
+ +
+ + + + + + +
+
+ + + +
+
+
+ + + + + +
+
+
+ + + +
+
+
+ + + +
+
+
+ + + +
+ +
+ + + + + +

fdtd

+ + +
+ + + +

+ meanas.fdtd + + +

+ +
+ +

Utilities for running finite-difference time-domain (FDTD) simulations

+

See the discussion of Maxwell's Equations in meanas.fdmath for basic +mathematical background.

+

Timestep

+

From the discussion of "Plane waves and the Dispersion relation" in meanas.fdmath, +we have

+
\[ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) \]
+

or, if \(\Delta_x = \Delta_y = \Delta_z\), then \(c \Delta_t < \frac{\Delta_x}{\sqrt{3}}\).

+

Based on this, we can set

+
dt = sqrt(mu.min() * epsilon.min()) / sqrt(1/dx_min**2 + 1/dy_min**2 + 1/dz_min**2)
+
+

The dx_min, dy_min, dz_min should be the minimum value across both the base and derived grids.

+

Poynting Vector and Energy Conservation

+

Let

+
\[ \begin{aligned} + \tilde{S}_{l, l', \vec{r}} &=& &\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}} \\ + &=& &\vec{x} (\tilde{E}^y_{l,m+1,n,p} \hat{H}^z_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^z_{l,m+1,n,p} \hat{H}^y_{l', \vec{r} + \frac{1}{2}}) \\ + & &+ &\vec{y} (\tilde{E}^z_{l,m,n+1,p} \hat{H}^x_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^x_{l,m,n+1,p} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) \\ + & &+ &\vec{z} (\tilde{E}^x_{l,m,n,p+1} \hat{H}^y_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^y_{l,m,n,p+1} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) + \end{aligned} +\]
+

where \(\vec{r} = (m, n, p)\) and \(\otimes\) is a modified cross product +in which the \(\tilde{E}\) terms are shifted as indicated.

+

By taking the divergence and rearranging terms, we can show that

+
\[ + \begin{aligned} + \hat{\nabla} \cdot \tilde{S}_{l, l', \vec{r}} + &= \hat{\nabla} \cdot (\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}}) \\ + &= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l, \vec{r}} - + \tilde{E}_{l, \vec{r}} \cdot \hat{\nabla} \times \hat{H}_{l', \vec{r} + \frac{1}{2}} \\ + &= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot + (-\tilde{\partial}_t \mu_{\vec{r} + \frac{1}{2}} \hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} - + \hat{M}_{l, \vec{r} + \frac{1}{2}}) - + \tilde{E}_{l, \vec{r}} \cdot (\hat{\partial}_t \tilde{\epsilon}_{\vec{r}} \tilde{E}_{l'+\frac{1}{2}, \vec{r}} + + \tilde{J}_{l', \vec{r}}) \\ + &= \hat{H}_{l'} \cdot (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) - + \tilde{E}_l \cdot (\epsilon / \Delta_t )(\tilde{E}_{l'+\frac{1}{2}} - \tilde{E}_{l'-\frac{1}{2}}) + - \hat{H}_{l'} \cdot \hat{M}_{l} - \tilde{E}_l \cdot \tilde{J}_{l'} \\ + \end{aligned} +\]
+

where in the last line the spatial subscripts have been dropped to emphasize +the time subscripts \(l, l'\), i.e.

+
\[ + \begin{aligned} + \tilde{E}_l &= \tilde{E}_{l, \vec{r}} \\ + \hat{H}_l &= \tilde{H}_{l, \vec{r} + \frac{1}{2}} \\ + \tilde{\epsilon} &= \tilde{\epsilon}_{\vec{r}} \\ + \end{aligned} +\]
+

etc. +For \(l' = l + \frac{1}{2}\) we get

+
\[ + \begin{aligned} + \hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} + &= \hat{H}_{l + \frac{1}{2}} \cdot + (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) - + \tilde{E}_l \cdot (\epsilon / \Delta_t)(\tilde{E}_{l+1} - \tilde{E}_l) + - \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\ + &= (-\mu / \Delta_t)(\hat{H}^2_{l + \frac{1}{2}} - \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}) - + (\epsilon / \Delta_t)(\tilde{E}_{l+1} \cdot \tilde{E}_l - \tilde{E}^2_l) + - \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\ + &= -(\mu \hat{H}^2_{l + \frac{1}{2}} + +\epsilon \tilde{E}_{l+1} \cdot \tilde{E}_l) / \Delta_t \\ + +(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} + +\epsilon \tilde{E}^2_l) / \Delta_t \\ + - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\ + \end{aligned} +\]
+

and for \(l' = l - \frac{1}{2}\),

+
\[ + \begin{aligned} + \hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} + &= (\mu \hat{H}^2_{l - \frac{1}{2}} + +\epsilon \tilde{E}_{l-1} \cdot \tilde{E}_l) / \Delta_t \\ + -(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} + +\epsilon \tilde{E}^2_l) / \Delta_t \\ + - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\ + \end{aligned} +\]
+

These two results form the discrete time-domain analogue to Poynting's theorem. +They hint at the expressions for the energy, which can be calculated at the same +time-index as either the E or H field:

+
\[ + \begin{aligned} + U_l &= \epsilon \tilde{E}^2_l + \mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} \\ + U_{l + \frac{1}{2}} &= \epsilon \tilde{E}_l \cdot \tilde{E}_{l + 1} + \mu \hat{H}^2_{l + \frac{1}{2}} \\ + \end{aligned} +\]
+

Rewriting the Poynting theorem in terms of the energy expressions,

+
\[ + \begin{aligned} + (U_{l+\frac{1}{2}} - U_l) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\ + - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\ + (U_l - U_{l-\frac{1}{2}}) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\ + - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\ + \end{aligned} +\]
+

This result is exact and should practically hold to within numerical precision. No time- +or spatial-averaging is necessary.

+

Note that each value of \(J\) contributes to the energy twice (i.e. once per field update) +despite only causing the value of \(E\) to change once (same for \(M\) and \(H\)).

+

Sources

+

It is often useful to excite the simulation with an arbitrary broadband pulse and then +extract the frequency-domain response by performing an on-the-fly Fourier transform +of the time-domain fields.

+

accumulate_phasor in meanas.fdtd.phasor performs the phase accumulation for one +or more target frequencies, while leaving source normalization and simulation-loop +policy to the caller. Convenience wrappers accumulate_phasor_e, +accumulate_phasor_h, and accumulate_phasor_j apply the usual Yee time offsets. +The helpers accumulate

+
\[ \Delta_t \sum_l w_l e^{-i \omega t_l} f_l \]
+

with caller-provided sample times and weights. In this codebase the matching +electric-current convention is typically E -= dt * J / epsilon in FDTD and +-i \omega J on the right-hand side of the FDFD wave equation.

+

The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse +shape. It can be written

+
\[ f_r(t) = (1 - \frac{1}{2} (\omega (t - \tau))^2) e^{-(\frac{\omega (t - \tau)}{2})^2} \]
+

with \(\tau > \frac{2 * \pi}{\omega}\) as a minimum delay to avoid a discontinuity at +t=0 (assuming the source is off for t<0 this gives \(\sim 10^{-3}\) error at t=0).

+

Boundary conditions

+

meanas.fdtd exposes two boundary-related building blocks:

+
    +
  • conducting_boundary(...) for simple perfect-electric-conductor style field + clamping at one face of the domain.
    • +
  • cpml_params(...) and updates_with_cpml(...) for convolutional perfectly + matched layers (CPMLs) attached to one or more faces of the Yee grid.
    • +
+

updates_with_cpml(...) accepts a three-by-two table of CPML parameter blocks:

+
cpml_params[axis][polarity_index]
+
+

where axis is 0, 1, or 2 and polarity_index corresponds to (-1, +1). +Passing None for one entry disables CPML on that face while leaving the other +directions unchanged. This is how mixed boundary setups such as "absorbing in x, +periodic in y/z" are expressed.

+

When comparing an FDTD run against an FDFD solve, use the same stretched +coordinate system in both places:

+
    +
  1. Build the FDTD update with the desired CPML parameters.
    2. +
  2. Stretch the FDFD dxes with the matching SCPML transform.
    3. +
  3. Compare the extracted phasor against the FDFD field or residual on those + stretched dxes.
    4. +
+

The electric-current sign convention used throughout the examples and tests is

+
\[ +E \leftarrow E - \Delta_t J / \epsilon +\]
+

which matches the FDFD right-hand side

+
\[ +-i \omega J. +\]
+ + + + + + + + + + +
+ + + + + + + + + + + + +
+ +
+ +

Core update and analysis helpers

+ + +
+ + + +

+ meanas.fdtd.base + + +

+ +
+ +

Basic FDTD field updates

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ maxwell_e + + +

+
maxwell_e(
+    dt: float, dxes: dx_lists_t | None = None
+) -> fdfield_updater_t
+
+ +
+ +

Build a function which performs a portion the time-domain E-field update,

+
E += curl_back(H[t]) / epsilon
+
+

The full update should be

+
E += (curl_back(H[t]) + J) / epsilon
+
+

which requires an additional step of E += J / epsilon which is not performed +by the generated function.

+

See meanas.fdmath for descriptions of

+
    +
  • This update step: "Maxwell's equations" section
    • +
  • dxes: "Datastructure: dx_lists_t" section
    • +
  • epsilon: "Permittivity and Permeability" section
    • +
+

Also see the "Timestep" section of meanas.fdtd for a discussion of +the dt parameter.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dt + + float + +
+

Timestep. See meanas.fdtd for details.

+
+ 		+ required +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_updater_t + +
+

Function f(E_old, H_old, epsilon) -> E_new.

+
+
+ + +
+ +
+ +
+ + +

+ maxwell_h + + +

+
maxwell_h(
+    dt: float, dxes: dx_lists_t | None = None
+) -> fdfield_updater_t
+
+ +
+ +

Build a function which performs part of the time-domain H-field update,

+
H -= curl_forward(E[t]) / mu
+
+

The full update should be

+
H -= (curl_forward(E[t]) + M) / mu
+
+

which requires an additional step of H -= M / mu which is not performed +by the generated function; this step can be omitted if there is no magnetic +current M.

+

See meanas.fdmath for descriptions of

+
    +
  • This update step: "Maxwell's equations" section
    • +
  • dxes: "Datastructure: dx_lists_t" section
    • +
  • mu: "Permittivity and Permeability" section
    • +
+

Also see the "Timestep" section of meanas.fdtd for a discussion of +the dt parameter.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dt + + float + +
+

Timestep. See meanas.fdtd for details.

+
+ 		+ required +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_updater_t + +
+

Function f(E_old, H_old, epsilon) -> E_new.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdtd.pml + + +

+ +
+ +

Convolutional perfectly matched layer (CPML) support for FDTD updates.

+

The helpers in this module construct per-face CPML parameters and then wrap the +standard Yee updates with the additional auxiliary psi fields needed by the +CPML recurrence.

+

The intended call pattern is:

+
    +
  1. Build a cpml_params[axis][polarity_index] table with cpml_params(...).
    2. +
  2. Pass that table into updates_with_cpml(...) together with dt, dxes, and + epsilon.
    3. +
  3. Advance the returned update_E / update_H closures in the simulation loop.
    4. +
+

Each face can be enabled or disabled independently by replacing one table entry +with None.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ cpml_params + + +

+
cpml_params(
+    axis: int,
+    polarity: int,
+    dt: float,
+    thickness: int = 8,
+    ln_R_per_layer: float = -1.6,
+    epsilon_eff: float = 1,
+    mu_eff: float = 1,
+    m: float = 3.5,
+    ma: float = 1,
+    cfs_alpha: float = 0,
+) -> dict[str, Any]
+
+ +
+ +

Construct the parameter block for one CPML face.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ axis + + int + +
+

Which Cartesian axis the CPML is normal to (0, 1, or 2).

+
+ 		+ required +
+ polarity + + int + +
+

Which face along that axis (-1 for the low-index face, ++1 for the high-index face).

+
+ 		+ required +
+ dt + + float + +
+

Timestep used by the Yee update.

+
+ 		+ required +
+ thickness + + int + +
+

Number of Yee cells occupied by the CPML region.

+
+ 		+ 8 +
+ ln_R_per_layer + + float + +
+

Logarithmic attenuation target per layer.

+
+ 		+ -1.6 +
+ epsilon_eff + + float + +
+

Effective permittivity used when choosing the CPML scaling.

+
+ 		+ 1 +
+ mu_eff + + float + +
+

Effective permeability used when choosing the CPML scaling.

+
+ 		+ 1 +
+ m + + float + +
+

Polynomial grading exponent for sigma and kappa.

+
+ 		+ 3.5 +
+ ma + + float + +
+

Polynomial grading exponent for the complex-frequency shift alpha.

+
+ 		+ 1 +
+ cfs_alpha + + float + +
+

Maximum complex-frequency shift parameter.

+
+ 		+ 0 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + + + + + + + + + +
TypeDescription
+ dict[str, Any] + +
+

Dictionary with:

+
+
+ dict[str, Any] + +
+
    +
  • param_e: (p0, p1, p2) arrays for the E update
    • +
+
+
+ dict[str, Any] + +
+
    +
  • param_h: (p0, p1, p2) arrays for the H update
    • +
+
+
+ dict[str, Any] + +
+
    +
  • region: slice tuple selecting the CPML cells on that face
    • +
+
+
+ + +
+ +
+ +
+ + +

+ updates_with_cpml + + +

+
updates_with_cpml(
+    cpml_params: Sequence[Sequence[dict[str, Any] | None]],
+    dt: float,
+    dxes: dx_lists_t,
+    epsilon: fdfield,
+    *,
+    dtype: DTypeLike = numpy.float32,
+) -> tuple[
+    Callable[[fdfield_t, fdfield_t, fdfield_t], None],
+    Callable[[fdfield_t, fdfield_t, fdfield_t], None],
+]
+
+ +
+ +

Build Yee-step update closures augmented with CPML terms.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ cpml_params + + Sequence[Sequence[dict[str, Any] | None]] + +
+

Three-by-two sequence indexed as [axis][polarity_index]. +Entries are the dictionaries returned by cpml_params(...); use +None to disable CPML on one face.

+
+ 		+ required +
+ dt + + float + +
+

Timestep.

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Yee-grid spacing lists [dx_e, dx_h].

+
+ 		+ required +
+ epsilon + + fdfield + +
+

Electric material distribution used by the E update.

+
+ 		+ required +
+ dtype + + DTypeLike + +
+

Storage dtype for the auxiliary CPML state arrays.

+
+ 		+ float32 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
TypeDescription
+ Callable[[fdfield_t, fdfield_t, fdfield_t], None] + +
+

(update_E, update_H) closures with the same call shape as the basic

+
+
+ Callable[[fdfield_t, fdfield_t, fdfield_t], None] + +
+

Yee updates:

+
+
+ tuple[Callable[[fdfield_t, fdfield_t, fdfield_t], None], Callable[[fdfield_t, fdfield_t, fdfield_t], None]] + +
+
    +
  • update_E(e, h, epsilon)
    • +
+
+
+ tuple[Callable[[fdfield_t, fdfield_t, fdfield_t], None], Callable[[fdfield_t, fdfield_t, fdfield_t], None]] + +
+
    +
  • update_H(e, h, mu)
    • +
+
+
+ tuple[Callable[[fdfield_t, fdfield_t, fdfield_t], None], Callable[[fdfield_t, fdfield_t, fdfield_t], None]] + +
+

The closures retain the CPML auxiliary state internally.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdtd.boundaries + + +

+ +
+ +

Boundary conditions

+

TODO conducting boundary documentation

+ + + + + + + + + + +
+ + + + + + + + + + + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdtd.energy + + +

+ +
+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ poynting + + +

+
poynting(
+    e: fdfield, h: fdfield, dxes: dx_lists_t | None = None
+) -> fdfield_t
+
+ +
+ +

Calculate the poynting vector S (\(S\)).

+

This is the energy transfer rate (amount of energy U per dt transferred +between adjacent cells) in each direction that happens during the half-step +bounded by the two provided fields.

+

The returned vector field S is the energy flow across +x, +y, and +z +boundaries of the corresponding U cell. For example,

+
    mx = numpy.roll(mask, -1, axis=0)
+    my = numpy.roll(mask, -1, axis=1)
+    mz = numpy.roll(mask, -1, axis=2)
+
+    u_hstep = fdtd.energy_hstep(e0=es[ii - 1], h1=hs[ii], e2=es[ii],     **args)
+    u_estep = fdtd.energy_estep(h0=hs[ii],     e1=es[ii], h2=hs[ii + 1], **args)
+    delta_j_B = fdtd.delta_energy_j(j0=js[ii], e1=es[ii], dxes=dxes)
+    du_half_h2e = u_estep - u_hstep - delta_j_B
+
+    s_h2e = -fdtd.poynting(e=es[ii], h=hs[ii], dxes=dxes) * dt
+    planes = [s_h2e[0, mask].sum(), -s_h2e[0, mx].sum(),
+              s_h2e[1, mask].sum(), -s_h2e[1, my].sum(),
+              s_h2e[2, mask].sum(), -s_h2e[2, mz].sum()]
+
+    assert_close(sum(planes), du_half_h2e[mask])
+
+

(see meanas.tests.test_fdtd.test_poynting_planes)

+

The full relationship is +$$ + \begin{aligned} + (U_{l+\frac{1}{2}} - U_l) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}{l, l + \frac{1}{2}} \ + - \hat{H}}{2}} \cdot \hat{Ml \ + - \tilde{E}_l \cdot \tilde{J} \ + (U_l - U_{l-\frac{1}{2}}) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}}{2}{l, l - \frac{1}{2}} \ + - \hat{H}}{2}} \cdot \hat{Ml \ + - \tilde{E}_l \cdot \tilde{J} \ + \end{aligned} +$$}{2}

+

These equalities are exact and should practically hold to within numerical precision. +No time- or spatial-averaging is necessary. (See meanas.fdtd docs for derivation.)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e + + fdfield + +
+

E-field

+
+ 		+ required +
+ h + + fdfield + +
+

H-field (one half-timestep before or after e)

+
+ 		+ required +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
Name TypeDescription
s + fdfield_t + +
+

Vector field. Components indicate the energy transfer rate from the +corresponding energy cell into its +x, +y, and +z neighbors during +the half-step from the time of the earlier input field until the +time of later input field.

+
+
+ + +
+ +
+ +
+ + +

+ poynting_divergence + + +

+
poynting_divergence(
+    s: fdfield | None = None,
+    *,
+    e: fdfield | None = None,
+    h: fdfield | None = None,
+    dxes: dx_lists_t | None = None,
+) -> fdfield_t
+
+ +
+ +

Calculate the divergence of the poynting vector.

+

This is the net energy flow for each cell, i.e. the change in energy U +per dt caused by transfer of energy to nearby cells (rather than +absorption/emission by currents J or M).

+

See poynting and meanas.fdtd for more details. +Args: + s: Poynting vector, as calculated with poynting. Optional; caller + can provide e and h instead. + e: E-field (optional; need either s or both e and h) + h: H-field (optional; need either s or both e and h) + dxes: Grid description; see meanas.fdmath.

+ + +

Returns:

+ + + + + + + + + + + + + +
Name TypeDescription
ds + fdfield_t + +
+

Divergence of the poynting vector. +Entries indicate the net energy flow out of the corresponding +energy cell.

+
+
+ + +
+ +
+ +
+ + +

+ energy_hstep + + +

+
energy_hstep(
+    e0: fdfield,
+    h1: fdfield,
+    e2: fdfield,
+    epsilon: fdfield | None = None,
+    mu: fdfield | None = None,
+    dxes: dx_lists_t | None = None,
+) -> fdfield_t
+
+ +
+ +

Calculate energy U at the time of the provided H-field h1.

+

TODO: Figure out what this means spatially.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e0 + + fdfield + +
+

E-field one half-timestep before the energy.

+
+ 		+ required +
+ h1 + + fdfield + +
+

H-field (at the same timestep as the energy).

+
+ 		+ required +
+ e2 + + fdfield + +
+

E-field one half-timestep after the energy.

+
+ 		+ required +
+ epsilon + + fdfield | None + +
+

Dielectric constant distribution.

+
+ 		+ None +
+ mu + + fdfield | None + +
+

Magnetic permeability distribution.

+
+ 		+ None +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_t + +
+

Energy, at the time of the H-field h1.

+
+
+ + +
+ +
+ +
+ + +

+ energy_estep + + +

+
energy_estep(
+    h0: fdfield,
+    e1: fdfield,
+    h2: fdfield,
+    epsilon: fdfield | None = None,
+    mu: fdfield | None = None,
+    dxes: dx_lists_t | None = None,
+) -> fdfield_t
+
+ +
+ +

Calculate energy U at the time of the provided E-field e1.

+

TODO: Figure out what this means spatially.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ h0 + + fdfield + +
+

H-field one half-timestep before the energy.

+
+ 		+ required +
+ e1 + + fdfield + +
+

E-field (at the same timestep as the energy).

+
+ 		+ required +
+ h2 + + fdfield + +
+

H-field one half-timestep after the energy.

+
+ 		+ required +
+ epsilon + + fdfield | None + +
+

Dielectric constant distribution.

+
+ 		+ None +
+ mu + + fdfield | None + +
+

Magnetic permeability distribution.

+
+ 		+ None +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_t + +
+

Energy, at the time of the E-field e1.

+
+
+ + +
+ +
+ +
+ + +

+ delta_energy_h2e + + +

+
delta_energy_h2e(
+    dt: float,
+    e0: fdfield,
+    h1: fdfield,
+    e2: fdfield,
+    h3: fdfield,
+    epsilon: fdfield | None = None,
+    mu: fdfield | None = None,
+    dxes: dx_lists_t | None = None,
+) -> fdfield_t
+
+ +
+ +

Change in energy during the half-step from h1 to e2.

+

This is just from (e2 * e2 + h3 * h1) - (h1 * h1 + e0 * e2)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e0 + + fdfield + +
+

E-field one half-timestep before the start of the energy delta.

+
+ 		+ required +
+ h1 + + fdfield + +
+

H-field at the start of the energy delta.

+
+ 		+ required +
+ e2 + + fdfield + +
+

E-field at the end of the energy delta (one half-timestep after h1).

+
+ 		+ required +
+ h3 + + fdfield + +
+

H-field one half-timestep after the end of the energy delta.

+
+ 		+ required +
+ epsilon + + fdfield | None + +
+

Dielectric constant distribution.

+
+ 		+ None +
+ mu + + fdfield | None + +
+

Magnetic permeability distribution.

+
+ 		+ None +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_t + +
+

Change in energy from the time of h1 to the time of e2.

+
+
+ + +
+ +
+ +
+ + +

+ delta_energy_e2h + + +

+
delta_energy_e2h(
+    dt: float,
+    h0: fdfield,
+    e1: fdfield,
+    h2: fdfield,
+    e3: fdfield,
+    epsilon: fdfield | None = None,
+    mu: fdfield | None = None,
+    dxes: dx_lists_t | None = None,
+) -> fdfield_t
+
+ +
+ +

Change in energy during the half-step from e1 to h2.

+

This is just from (h2 * h2 + e3 * e1) - (e1 * e1 + h0 * h2)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ h0 + + fdfield + +
+

E-field one half-timestep before the start of the energy delta.

+
+ 		+ required +
+ e1 + + fdfield + +
+

H-field at the start of the energy delta.

+
+ 		+ required +
+ h2 + + fdfield + +
+

E-field at the end of the energy delta (one half-timestep after e1).

+
+ 		+ required +
+ e3 + + fdfield + +
+

H-field one half-timestep after the end of the energy delta.

+
+ 		+ required +
+ epsilon + + fdfield | None + +
+

Dielectric constant distribution.

+
+ 		+ None +
+ mu + + fdfield | None + +
+

Magnetic permeability distribution.

+
+ 		+ None +
+ dxes + + dx_lists_t | None + +
+

Grid description; see meanas.fdmath.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_t + +
+

Change in energy from the time of e1 to the time of h2.

+
+
+ + +
+ +
+ +
+ + +

+ delta_energy_j + + +

+
delta_energy_j(
+    j0: fdfield, e1: fdfield, dxes: dx_lists_t | None = None
+) -> fdfield_t
+
+ +
+ +

Calculate the electric-current work term \(J \cdot E\) on the Yee grid.

+

This is the source contribution that appears beside the flux divergence in +the discrete Poynting identities documented in meanas.fdtd.

+

Note that each value of J contributes twice in a full Yee cycle (once per +half-step energy balance) even though it directly changes E only once.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ j0 + + fdfield + +
+

Electric-current density sampled at the same half-step as the +current work term.

+
+ 		+ required +
+ e1 + + fdfield + +
+

Electric field sampled at the matching integer timestep.

+
+ 		+ required +
+ dxes + + dx_lists_t | None + +
+

Grid description; defaults to unit spacing.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ fdfield_t + +
+

Per-cell source-work contribution with the scalar field shape.

+
+
+ + +
+ +
+ +
+ + +

+ dxmul + + +

+
dxmul(
+    ee: fdfield,
+    hh: fdfield,
+    epsilon: fdfield | float | None = None,
+    mu: fdfield | float | None = None,
+    dxes: dx_lists_t | None = None,
+) -> fdfield_t
+
+ +
+ +

Multiply E- and H-like field products by material weights and cell volumes.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ ee + + fdfield + +
+

Three-component electric-field product, such as e0 * e2.

+
+ 		+ required +
+ hh + + fdfield + +
+

Three-component magnetic-field product, such as h1 * h1.

+
+ 		+ required +
+ epsilon + + fdfield | float | None + +
+

Electric material weight; defaults to 1.

+
+ 		+ None +
+ mu + + fdfield | float | None + +
+

Magnetic material weight; defaults to 1.

+
+ 		+ None +
+ dxes + + dx_lists_t | None + +
+

Grid description; defaults to unit spacing.

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ fdfield_t + +
+

Scalar field containing the weighted electric plus magnetic contribution

+
+
+ fdfield_t + +
+

for each Yee cell.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdtd.phasor + + +

+ +
+ +

Helpers for extracting single- or multi-frequency phasors from FDTD samples.

+

These helpers are intentionally low-level: callers own the accumulator arrays and +the sampling policy. The accumulated quantity is

+
dt * sum(weight * exp(-1j * omega * t_step) * sample_step)
+
+

where t_step = (step + offset_steps) * dt.

+

The usual Yee offsets are:

+
    +
  • accumulate_phasor_e(..., step=l) for E_l
    • +
  • accumulate_phasor_h(..., step=l) for H_{l + 1/2}
    • +
  • accumulate_phasor_j(..., step=l) for J_{l + 1/2}
    • +
+

These helpers do not choose warmup/accumulation windows or pulse-envelope +normalization. They also do not impose a current sign convention. In this +codebase, electric-current injection normally appears as E -= dt * J / epsilon, +which matches the FDFD right-hand side -1j * omega * J.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ accumulate_phasor + + +

+
accumulate_phasor(
+    accumulator: NDArray[complexfloating],
+    omegas: float
+    | complex
+    | Sequence[float | complex]
+    | NDArray,
+    dt: float,
+    sample: ArrayLike,
+    step: int,
+    *,
+    offset_steps: float = 0.0,
+    weight: ArrayLike = 1.0,
+) -> NDArray[numpy.complexfloating]
+
+ +
+ +

Add one time-domain sample into a phasor accumulator.

+

The added quantity is

+
dt * weight * exp(-1j * omega * t_step) * sample
+
+

where t_step = (step + offset_steps) * dt.

+ + +
+ Note +

This helper already multiplies by dt. If the caller's normalization +factor was derived from a discrete sum that already includes dt, pass +weight / dt here.

+
+ +
+ +
+ +
+ + +

+ accumulate_phasor_e + + +

+
accumulate_phasor_e(
+    accumulator: NDArray[complexfloating],
+    omegas: float
+    | complex
+    | Sequence[float | complex]
+    | NDArray,
+    dt: float,
+    sample: ArrayLike,
+    step: int,
+    *,
+    weight: ArrayLike = 1.0,
+) -> NDArray[numpy.complexfloating]
+
+ +
+ +

Accumulate an E-field sample taken at integer timestep step.

+ + +
+ +
+ +
+ + +

+ accumulate_phasor_h + + +

+
accumulate_phasor_h(
+    accumulator: NDArray[complexfloating],
+    omegas: float
+    | complex
+    | Sequence[float | complex]
+    | NDArray,
+    dt: float,
+    sample: ArrayLike,
+    step: int,
+    *,
+    weight: ArrayLike = 1.0,
+) -> NDArray[numpy.complexfloating]
+
+ +
+ +

Accumulate an H-field sample corresponding to H_{step + 1/2}.

+ + +
+ +
+ +
+ + +

+ accumulate_phasor_j + + +

+
accumulate_phasor_j(
+    accumulator: NDArray[complexfloating],
+    omegas: float
+    | complex
+    | Sequence[float | complex]
+    | NDArray,
+    dt: float,
+    sample: ArrayLike,
+    step: int,
+    *,
+    weight: ArrayLike = 1.0,
+) -> NDArray[numpy.complexfloating]
+
+ +
+ +

Accumulate a current sample corresponding to J_{step + 1/2}.

+ + +
+ +
+ + + +
+ +
+ +
+ + + + + + + + + + + + + +
+
+ + + +
+ + + +
+ + + +
+
+
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API Overview

+

The package is documented directly from its docstrings. The most useful entry +points are:

+
    +
  • meanas: top-level package overview
    • +
  • eigensolvers: generic eigenvalue utilities used by the mode solvers
    • +
  • fdfd: frequency-domain operators, sources, PML, solvers, and far-field transforms
    • +
  • waveguides: straight, cylindrical, and 3D waveguide mode helpers
    • +
  • fdtd: timestepping, CPML, energy/flux helpers, and phasor extraction
    • +
  • fdmath: shared discrete operators, vectorization helpers, and derivation background
    • +
+

The waveguide and FDTD pages are the best places to start if you want the +mathematical derivations rather than just the callable reference.

+ + + + + + + + + + + + + +
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meanas

+ + +
+ + + +

+ meanas + + +

+ +
+ +

Electromagnetic simulation tools

+

See the readme or import meanas; help(meanas) for more info.

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waveguides

+ + +
+ + + +

+ meanas.fdfd.waveguide_2d + + +

+ +
+ +

Operators and helper functions for waveguides with unchanging cross-section.

+

The propagation direction is chosen to be along the z axis, and all fields +are given an implicit z-dependence of the form exp(-1 * wavenumber * z).

+

As the z-dependence is known, all the functions in this file assume a 2D grid + (i.e. dxes = [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]], [[dx_h[0], ...], [dy_h[0], ...]]]).

+

===============

+

Consider Maxwell's equations in continuous space, in the frequency domain. Assuming +a structure with some (x, y) cross-section extending uniformly into the z dimension, +with a diagonal \(\epsilon\) tensor, we have

+
\[ +\begin{aligned} +\nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\ +\nabla \times \vec{H}(x, y, z) &= \imath \omega \epsilon \vec{E} \\ +\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\imath \beta z} \\ +\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\imath \beta z} \\ +\end{aligned} +\]
+

Expanding the first two equations into vector components, we get

+
\[ +\begin{aligned} +-\imath \omega \mu_{xx} H_x &= \partial_y E_z - \partial_z E_y \\ +-\imath \omega \mu_{yy} H_y &= \partial_z E_x - \partial_x E_z \\ +-\imath \omega \mu_{zz} H_z &= \partial_x E_y - \partial_y E_x \\ +\imath \omega \epsilon_{xx} E_x &= \partial_y H_z - \partial_z H_y \\ +\imath \omega \epsilon_{yy} E_y &= \partial_z H_x - \partial_x H_z \\ +\imath \omega \epsilon_{zz} E_z &= \partial_x H_y - \partial_y H_x \\ +\end{aligned} +\]
+

Substituting in our expressions for \(\vec{E}\), \(\vec{H}\) and discretizing:

+
\[ +\begin{aligned} +-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \imath \beta E_y \\ +-\imath \omega \mu_{yy} H_y &= -\imath \beta E_x - \tilde{\partial}_x E_z \\ +-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\ +\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta H_y \\ +\imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \hat{\partial}_x H_z \\ +\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\ +\end{aligned} +\]
+

Rewrite the last three equations as

+
\[ +\begin{aligned} +\imath \beta H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\ +\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\ +\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\ +\end{aligned} +\]
+

Now apply \(\imath \beta \tilde{\partial}_x\) to the last equation, +then substitute in for \(\imath \beta H_x\) and \(\imath \beta H_y\):

+
\[ +\begin{aligned} +\imath \beta \tilde{\partial}_x \imath \omega E_z &= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y + - \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\ + &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z) + - \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\ + &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x) + - \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y) \\ +\imath \beta \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x) + + \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\ +\end{aligned} +\]
+

With a similar approach (but using \(\imath \beta \tilde{\partial}_y\) instead), we can get

+
\[ +\begin{aligned} +\imath \beta \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x) + + \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\ +\end{aligned} +\]
+

We can combine this equation for \(\imath \beta \tilde{\partial}_y E_z\) with +the unused \(\imath \omega \mu_{xx} H_x\) and \(\imath \omega \mu_{yy} H_y\) equations to get

+
\[ +\begin{aligned} +-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \imath \beta \tilde{\partial}_y E_z \\ +-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \tilde{\partial}_y ( + \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x) + + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) + )\\ +\end{aligned} +\]
+

and

+
\[ +\begin{aligned} +-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \imath \beta \tilde{\partial}_x E_z \\ +-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \tilde{\partial}_x ( + \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x) + + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) + )\\ +\end{aligned} +\]
+

However, based on our rewritten equation for \(\imath \beta H_x\) and the so-far unused +equation for \(\imath \omega \mu_{zz} H_z\) we can also write

+
\[ +\begin{aligned} +-\imath \omega \mu_{xx} (\imath \beta H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\ + &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + \imath \omega \mu_{xx} \hat{\partial}_x ( + \frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\ + &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + -\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\ +\end{aligned} +\]
+

and, similarly,

+
\[ +\begin{aligned} +-\imath \omega \mu_{yy} (\imath \beta H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x + +\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\ +\end{aligned} +\]
+

By combining both pairs of expressions, we get

+
\[ +\begin{aligned} +\beta^2 E_x - \tilde{\partial}_x ( + \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x) + + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) + ) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x + +\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\ +-\beta^2 E_y + \tilde{\partial}_y ( + \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x) + + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) + ) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + -\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\ +\end{aligned} +\]
+

Using these, we can construct the eigenvalue problem

+
\[ +\beta^2 \begin{bmatrix} E_x \\ + E_y \end{bmatrix} = + (\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\ + 0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} + + \begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\ + \mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1} + \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} + + \begin{bmatrix} \tilde{\partial}_x \\ + \tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1} + \begin{bmatrix} \hat{\partial}_x \epsilon_{xx} & \hat{\partial}_y \epsilon_{yy} \end{bmatrix}) + \begin{bmatrix} E_x \\ + E_y \end{bmatrix} +\]
+

In the literature, \(\beta\) is usually used to denote the lossless/real part of the propagation constant, +but in meanas it is allowed to be complex.

+

An equivalent eigenvalue problem can be formed using the \(H_x\) and \(H_y\) fields, if those are more convenient.

+

Note that \(E_z\) was never discretized, so \(\beta\) will need adjustment to account for numerical dispersion +if the result is introduced into a space with a discretized z-axis.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ operator_e + + +

+
operator_e(
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Waveguide operator of the form

+
omega**2 * mu * epsilon +
+mu * [[-Dy], [Dx]] / mu * [-Dy, Dx] +
+[[Dx], [Dy]] / epsilon * [Dx, Dy] * epsilon
+
+

for use with a field vector of the form cat([E_x, E_y]).

+

More precisely, the operator is

+
\[ +\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\ + 0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} + + \begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\ + \mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1} + \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} + + \begin{bmatrix} \tilde{\partial}_x \\ + \tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1} + \begin{bmatrix} \hat{\partial}_x \epsilon_{xx} & \hat{\partial}_y \epsilon_{yy} \end{bmatrix} +\]
+

\(\tilde{\partial}_x\) and \(\hat{\partial}_x\) are the forward and backward derivatives along x, +and each \(\epsilon_{xx}\), \(\mu_{yy}\), etc. is a diagonal matrix containing the vectorized material +property distribution.

+

This operator can be used to form an eigenvalue problem of the form +operator_e(...) @ [E_x, E_y] = wavenumber**2 * [E_x, E_y]

+

which can then be solved for the eigenmodes of the system (an exp(-i * wavenumber * z) +z-dependence is assumed for the fields).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

The angular frequency of the system.

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ operator_h + + +

+
operator_h(
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Waveguide operator of the form

+
omega**2 * epsilon * mu +
+epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
+[[Dx], [Dy]] / mu * [Dx, Dy] * mu
+
+

for use with a field vector of the form cat([H_x, H_y]).

+

More precisely, the operator is

+
\[ +\omega^2 \begin{bmatrix} \epsilon_{yy} \mu_{xx} & 0 \\ + 0 & \epsilon_{xx} \mu_{yy} \end{bmatrix} + + \begin{bmatrix} -\epsilon_{yy} \tilde{\partial}_y \\ + \epsilon_{xx} \tilde{\partial}_x \end{bmatrix} \epsilon_{zz}^{-1} + \begin{bmatrix} -\hat{\partial}_y & \hat{\partial}_x \end{bmatrix} + + \begin{bmatrix} \hat{\partial}_x \\ + \hat{\partial}_y \end{bmatrix} \mu_{zz}^{-1} + \begin{bmatrix} \tilde{\partial}_x \mu_{xx} & \tilde{\partial}_y \mu_{yy} \end{bmatrix} +\]
+

\(\tilde{\partial}_x\) and \(\hat{\partial}_x\) are the forward and backward derivatives along x, +and each \(\epsilon_{xx}\), \(\mu_{yy}\), etc. is a diagonal matrix containing the vectorized material +property distribution.

+

This operator can be used to form an eigenvalue problem of the form +operator_h(...) @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]

+

which can then be solved for the eigenmodes of the system (an exp(-i * wavenumber * z) +z-dependence is assumed for the fields).

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + complex + +
+

The angular frequency of the system.

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ normalized_fields_e + + +

+
normalized_fields_e(
+    e_xy: vcfdfield2,
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+    prop_phase: float = 0,
+) -> tuple[vcfdslice_t, vcfdslice_t]
+
+ +
+ +

Given a vector e_xy containing the vectorized E_x and E_y fields, + returns normalized, vectorized E and H fields for the system.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e_xy + + vcfdfield2 + +
+

Vector containing E_x and E_y fields

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z). + It should satisfy operator_e() @ e_xy == wavenumber**2 * e_xy

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ prop_phase + + float + +
+

Phase shift (dz * corrected_wavenumber) over 1 cell in propagation direction. + Default 0 (continuous propagation direction, i.e. dz->0).

+
+ 		+ 0 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ vcfdslice_t + +
+

(e, h), where each field is vectorized, normalized,

+
+
+ vcfdslice_t + +
+

and contains all three vector components.

+
+
+ + +
+ Notes +

e_xy is only the transverse electric eigenvector. This helper first +reconstructs the full three-component E and H fields with exy2e(...) +and exy2h(...), then normalizes them to unit forward power using +_normalized_fields(...).

+

The normalization target is

+
\[ +\Re\left[\mathrm{inner\_product}(e, h, \mathrm{conj\_h}=True)\right] = 1, +\]
+

so the returned fields represent a unit-power forward mode under the +discrete Yee-grid Poynting inner product.

+
+ +
+ +
+ +
+ + +

+ normalized_fields_h + + +

+
normalized_fields_h(
+    h_xy: vcfdfield2,
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+    prop_phase: float = 0,
+) -> tuple[vcfdslice_t, vcfdslice_t]
+
+ +
+ +

Given a vector h_xy containing the vectorized H_x and H_y fields, + returns normalized, vectorized E and H fields for the system.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ h_xy + + vcfdfield2 + +
+

Vector containing H_x and H_y fields

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z). + It should satisfy operator_h() @ h_xy == wavenumber**2 * h_xy

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ prop_phase + + float + +
+

Phase shift (dz * corrected_wavenumber) over 1 cell in propagation direction. + Default 0 (continuous propagation direction, i.e. dz->0).

+
+ 		+ 0 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ vcfdslice_t + +
+

(e, h), where each field is vectorized, normalized,

+
+
+ vcfdslice_t + +
+

and contains all three vector components.

+
+
+ + +
+ Notes +

This is the H_x/H_y analogue of normalized_fields_e(...). The final +normalized mode should describe the same physical solution, but because +the overall complex phase and sign are chosen heuristically, +normalized_fields_e(...) and normalized_fields_h(...) need not return +identical representatives for nearly symmetric modes.

+
+ +
+ +
+ +
+ + +

+ exy2h + + +

+
exy2h(
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Operator which transforms the vector e_xy containing the vectorized E_x and E_y fields, + into a vectorized H containing all three H components

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z). + It should satisfy operator_e() @ e_xy == wavenumber**2 * e_xy

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representing the operator.

+
+
+ + +
+ +
+ +
+ + +

+ hxy2e + + +

+
hxy2e(
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Operator which transforms the vector h_xy containing the vectorized H_x and H_y fields, + into a vectorized E containing all three E components

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z). + It should satisfy operator_h() @ h_xy == wavenumber**2 * h_xy

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representing the operator.

+
+
+ + +
+ +
+ +
+ + +

+ hxy2h + + +

+
hxy2h(
+    wavenumber: complex,
+    dxes: dx_lists2_t,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Operator which transforms the vector h_xy containing the vectorized H_x and H_y fields, + into a vectorized H containing all three H components

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z). + It should satisfy operator_h() @ h_xy == wavenumber**2 * h_xy

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representing the operator.

+
+
+ + +
+ +
+ +
+ + +

+ exy2e + + +

+
exy2e(
+    wavenumber: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+) -> sparse.sparray
+
+ +
+ +

Operator which transforms the vector e_xy containing the vectorized E_x and E_y fields, + into a vectorized E containing all three E components

+

From the operator derivation (see module docs), we have

+
\[ +\imath \omega \epsilon_{zz} E_z = \hat{\partial}_x H_y - \hat{\partial}_y H_x \\ +\]
+

as well as the intermediate equations

+
\[ +\begin{aligned} +\imath \beta H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\ +\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\ +\end{aligned} +\]
+

Combining these, we get

+
\[ +\begin{aligned} +E_z &= \frac{1}{- \omega \beta \epsilon_{zz}} (( + \hat{\partial}_y \hat{\partial}_x H_z + -\hat{\partial}_x \hat{\partial}_y H_z) + + \imath \omega (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)) + &= \frac{1}{\imath \beta \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y) +\end{aligned} +\]
+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z) + It should satisfy operator_e() @ e_xy == wavenumber**2 * e_xy

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representing the operator.

+
+
+ + +
+ +
+ +
+ + +

+ e2h + + +

+
e2h(
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Returns an operator which, when applied to a vectorized E eigenfield, produces + the vectorized H eigenfield slice.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z)

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ h2e + + +

+
h2e(
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+) -> sparse.sparray
+
+ +
+ +

Returns an operator which, when applied to a vectorized H eigenfield, produces + the vectorized E eigenfield slice.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z)

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ curl_e + + +

+
curl_e(
+    wavenumber: complex, dxes: dx_lists2_t
+) -> sparse.sparray
+
+ +
+ +

Discretized curl operator for use with the waveguide E field slice.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z)

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ curl_h + + +

+
curl_h(
+    wavenumber: complex, dxes: dx_lists2_t
+) -> sparse.sparray
+
+ +
+ +

Discretized curl operator for use with the waveguide H field slice.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z)

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ h_err + + +

+
h_err(
+    h: vcfdslice,
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> float
+
+ +
+ +

Calculates the relative error in the H field

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ h + + vcfdslice + +
+

Vectorized H field

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z)

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ float + +
+

Relative error norm(A_h @ h) / norm(h).

+
+
+ + +
+ +
+ +
+ + +

+ e_err + + +

+
e_err(
+    e: vcfdslice,
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> float
+
+ +
+ +

Calculates the relative error in the E field

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e + + vcfdslice + +
+

Vectorized E field

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber assuming fields have z-dependence of exp(-i * wavenumber * z)

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ float + +
+

Relative error norm(A_e @ e) / norm(e).

+
+
+ + +
+ +
+ +
+ + +

+ sensitivity + + +

+
sensitivity(
+    e_norm: vcfdslice,
+    h_norm: vcfdslice,
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> vcfdslice_t
+
+ +
+ +

Given a waveguide structure (dxes, epsilon, mu) and mode fields +(e_norm, h_norm, wavenumber, omega), calculates the sensitivity of the wavenumber +\(\beta\) to changes in the dielectric structure \(\epsilon\).

+

The output is a vector of the same size as vec(epsilon), with each element specifying the +sensitivity of wavenumber to changes in the corresponding element in vec(epsilon), i.e.

+
\[sens_{i} = \frac{\partial\beta}{\partial\epsilon_i}\]
+

An adjoint approach is used to calculate the sensitivity; the derivation is provided here:

+

Starting with the eigenvalue equation

+
\[\beta^2 E_{xy} = A_E E_{xy}\]
+

where \(A_E\) is the waveguide operator from operator_e(), and \(E_{xy} = \begin{bmatrix} E_x \\ + E_y \end{bmatrix}\), +we can differentiate with respect to one of the \(\epsilon\) elements (i.e. at one Yee grid point), \(\epsilon_i\):

+
\[ +(2 \beta) \partial_{\epsilon_i}(\beta) E_{xy} + \beta^2 \partial_{\epsilon_i} E_{xy} + = \partial_{\epsilon_i}(A_E) E_{xy} + A_E \partial_{\epsilon_i} E_{xy} +\]
+

We then multiply by \(H_{yx}^\star = \begin{bmatrix}H_y^\star \\ -H_x^\star \end{bmatrix}\) from the left:

+
\[ +(2 \beta) \partial_{\epsilon_i}(\beta) H_{yx}^\star E_{xy} + \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy} + = H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} + H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy} +\]
+

However, \(H_{yx}^\star\) is actually a left-eigenvector of \(A_E\). This can be verified by inspecting +the form of operator_h (\(A_H\)) and comparing its conjugate transpose to operator_e (\(A_E\)). Also, note +\(H_{yx}^\star \cdot E_{xy} = H^\star \times E\) recalls the mode orthogonality relation. See doi:10.5194/ars-9-85-201 +for a similar approach. Therefore,

+
\[ +H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy} = \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy} +\]
+

and we can simplify to

+
\[ +\partial_{\epsilon_i}(\beta) + = \frac{1}{2 \beta} \frac{H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} }{H_{yx}^\star E_{xy}} +\]
+

This expression can be quickly calculated for all \(i\) by writing out the various terms of +\(\partial_{\epsilon_i} A_E\) and recognizing that the vector-matrix-vector products (i.e. scalars) +\(sens_i = \vec{v}_{left} \partial_{\epsilon_i} (\epsilon_{xyz}) \vec{v}_{right}\), indexed by \(i\), can be expressed as +elementwise multiplications \(\vec{sens} = \vec{v}_{left} \star \vec{v}_{right}\)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e_norm + + vcfdslice + +
+

Normalized, vectorized E_xyz field for the mode. E.g. as returned by normalized_fields_e.

+
+ 		+ required +
+ h_norm + + vcfdslice + +
+

Normalized, vectorized H_xyz field for the mode. E.g. as returned by normalized_fields_e.

+
+ 		+ required +
+ wavenumber + + complex + +
+

Propagation constant for the mode. The z-axis is assumed to be continuous (i.e. without numerical dispersion).

+
+ 		+ required +
+ omega + + complex + +
+

The angular frequency of the system.

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ vcfdslice_t + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ solve_modes + + +

+
solve_modes(
+    mode_numbers: Sequence[int],
+    omega: complex,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+    mode_margin: int = 2,
+) -> tuple[
+    NDArray[numpy.complex128], NDArray[numpy.complex128]
+]
+
+ +
+ +

Given a 2D region, attempts to solve for the eigenmode with the specified mode numbers.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ mode_numbers + + Sequence[int] + +
+

List of 0-indexed mode numbers to solve for

+
+ 		+ required +
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Dielectric constant

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ mode_margin + + int + +
+

The eigensolver will actually solve for (max(mode_number) + mode_margin) + modes, but only return the target mode. Increasing this value can improve the solver's + ability to find the correct mode. Default 2.

+
+ 		+ 2 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
Name TypeDescription
e_xys + NDArray[complex128] + +
+

NDArray of vfdfield_t specifying fields. First dimension is mode number.

+
+
wavenumbers + NDArray[complex128] + +
+

list of wavenumbers

+
+
+ + +
+ +
+ +
+ + +

+ solve_mode + + +

+
solve_mode(
+    mode_number: int, *args: Any, **kwargs: Any
+) -> tuple[vcfdfield2_t, complex]
+
+ +
+ +

Wrapper around solve_modes() that solves for a single mode.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ mode_number + + int + +
+

0-indexed mode number to solve for

+
+ 		+ required +
+ *args + + Any + +
+

passed to solve_modes()

+
+ 		+ () +
+ **kwargs + + Any + +
+

passed to solve_modes()

+
+ 		+ {} +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[vcfdfield2_t, complex] + +
+

(e_xy, wavenumber)

+
+
+ + +
+ +
+ +
+ + +

+ inner_product + + +

+
inner_product(
+    e1: vcfdfield2,
+    h2: vcfdfield2,
+    dxes: dx_lists2_t,
+    prop_phase: float = 0,
+    conj_h: bool = False,
+    trapezoid: bool = False,
+) -> complex
+
+ +
+ +

Compute the discrete waveguide overlap / Poynting inner product.

+

This is the 2D transverse integral corresponding to the time-averaged +longitudinal Poynting flux,

+
\[ +\frac{1}{2}\int (E_x H_y - E_y H_x) \, dx \, dy +\]
+

with the Yee-grid staggering and optional propagation-phase adjustment used +by the waveguide helpers in this module.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e1 + + vcfdfield2 + +
+

Vectorized electric field, typically from exy2e(...) or +normalized_fields_e(...).

+
+ 		+ required +
+ h2 + + vcfdfield2 + +
+

Vectorized magnetic field, typically from hxy2h(...), +exy2h(...), or one of the normalization helpers.

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Two-dimensional Yee-grid spacing lists [dx_e, dx_h].

+
+ 		+ required +
+ prop_phase + + float + +
+

Phase advance over one propagation cell. This is used to +shift the H field into the same longitudinal reference plane as the +E field.

+
+ 		+ 0 +
+ conj_h + + bool + +
+

Whether to conjugate h2 before forming the overlap. Use +True for the usual time-averaged power normalization.

+
+ 		+ False +
+ trapezoid + + bool + +
+

Whether to use trapezoidal quadrature instead of the default +rectangular Yee-cell sum.

+
+ 		+ False +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ complex + +
+

Complex overlap / longitudinal power integral.

+
+
+ + +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdfd.waveguide_3d + + +

+ +
+ +

Tools for working with waveguide modes in 3D domains.

+

This module relies heavily on waveguide_2d and mostly just transforms +its parameters into 2D equivalents and expands the results back into 3D.

+

The intended workflow is:

+
    +
  1. Select a single-cell slice normal to the propagation axis.
    2. +
  2. Solve the corresponding 2D mode problem with solve_mode(...).
    3. +
  3. Turn that mode into a one-sided source with compute_source(...).
    4. +
  4. Build an overlap window with compute_overlap_e(...) for port readout.
    5. +
+

polarity is part of the public convention throughout this module:

+
    +
  • +1 means forward propagation toward increasing index along axis
    • +
  • -1 means backward propagation toward decreasing index along axis
    • +
+

That same convention controls which side of the selected slice is used for the +overlap window and how the expanded field is phased.

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ solve_mode + + +

+
solve_mode(
+    mode_number: int,
+    omega: complex,
+    dxes: dx_lists_t,
+    axis: int,
+    polarity: int,
+    slices: Sequence[slice],
+    epsilon: fdfield,
+    mu: fdfield | None = None,
+) -> dict[str, complex | NDArray[complexfloating]]
+
+ +
+ +

Given a 3D grid, selects a slice from the grid and attempts to +solve for an eigenmode propagating through that slice.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ mode_number + + int + +
+

Number of the mode, 0-indexed

+
+ 		+ required +
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ axis + + int + +
+

Propagation axis (0=x, 1=y, 2=z)

+
+ 		+ required +
+ polarity + + int + +
+

Propagation direction (+1 for +ve, -1 for -ve)

+
+ 		+ required +
+ slices + + Sequence[slice] + +
+

epsilon[tuple(slices)] is used to select the portion of the grid to use + as the waveguide cross-section. slices[axis] must select exactly one item.

+
+ 		+ required +
+ epsilon + + fdfield + +
+

Dielectric constant

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
TypeDescription
+ dict[str, complex | NDArray[complexfloating]] + +
+

Dictionary containing:

+
+
+ dict[str, complex | NDArray[complexfloating]] + +
+
    +
  • E: full-grid electric field for the solved mode
    • +
+
+
+ dict[str, complex | NDArray[complexfloating]] + +
+
    +
  • H: full-grid magnetic field for the solved mode
    • +
+
+
+ dict[str, complex | NDArray[complexfloating]] + +
+
    +
  • wavenumber: propagation constant corrected for the discretized +propagation axis
    • +
+
+
+ dict[str, complex | NDArray[complexfloating]] + +
+
    +
  • wavenumber_2d: propagation constant of the reduced 2D eigenproblem
    • +
+
+
+ + +
+ Notes +

The returned fields are normalized through the waveguide_2d +normalization convention before being expanded back to 3D.

+
+ +
+ +
+ +
+ + +

+ compute_source + + +

+
compute_source(
+    E: cfdfield,
+    wavenumber: complex,
+    omega: complex,
+    dxes: dx_lists_t,
+    axis: int,
+    polarity: int,
+    slices: Sequence[slice],
+    epsilon: fdfield,
+    mu: fdfield | None = None,
+) -> cfdfield_t
+
+ +
+ +

Given an eigenmode obtained by solve_mode, returns the current source distribution +necessary to position a unidirectional source at the slice location.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ E + + cfdfield + +
+

E-field of the mode

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber of the mode

+
+ 		+ required +
+ omega + + complex + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ axis + + int + +
+

Propagation axis (0=x, 1=y, 2=z)

+
+ 		+ required +
+ polarity + + int + +
+

Propagation direction (+1 for +ve, -1 for -ve)

+
+ 		+ required +
+ slices + + Sequence[slice] + +
+

epsilon[tuple(slices)] is used to select the portion of the grid to use + as the waveguide cross-section. slices[axis] should select only one item.

+
+ 		+ required +
+ mu + + fdfield | None + +
+

Magnetic permeability (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ cfdfield_t + +
+

J distribution for a one-sided electric-current source.

+
+
+ + +
+ Notes +

The source is built from the expanded mode field and a boundary-source +operator. The resulting current is intended to be injected with the +same sign convention used elsewhere in the package:

+

E -= dt * J / epsilon

+
+ +
+ +
+ +
+ + +

+ compute_overlap_e + + +

+
compute_overlap_e(
+    E: cfdfield_t,
+    wavenumber: complex,
+    dxes: dx_lists_t,
+    axis: int,
+    polarity: int,
+    slices: Sequence[slice],
+    omega: float,
+) -> cfdfield_t
+
+ +
+ +

Build an overlap field for projecting another 3D electric field onto a mode.

+

The returned field is intended for the discrete overlap expression

+
\[ +\sum \mathrm{overlap\_e} \; E_\mathrm{other}^* +\]
+

where the sum is over the full Yee-grid field storage.

+

The construction uses a two-cell window immediately upstream of the selected +slice:

+
    +
  • for polarity=+1, the two cells just before slices[axis].start
    • +
  • for polarity=-1, the two cells just after slices[axis].stop
    • +
+

The window is clipped to the simulation domain if necessary. A clipped but +non-empty window raises RuntimeWarning; an empty window raises +ValueError.

+

The derivation below assumes reflection symmetry and the standard waveguide +overlap relation involving

+
\[ +\int ((E \times H_\mathrm{mode}) + (E_\mathrm{mode} \times H)) \cdot dn. +\]
+

E x H_mode + E_mode x H +-> Ex Hmy - EyHmx + Emx Hy - Emy Hx (Z-prop) +Ex we/B Emx + Ex i/B dy Hmz - Ey (-we/B Emy) - Ey i/B dx Hmz +we/B (Ex Emx + Ey Emy) + i/B (Ex dy Hmz - Ey dx Hmz) +we/B (Ex Emx + Ey Emy) + i/B (Ex dy (dx Emy - dy Emx) - Ey dx (dx Emy - dy Emx)) +we/B (Ex Emx + Ey Emy) + i/B (Ex dy dx Emy - Ex dy dy Emx - Ey dx dx Emy - Ey dx dy Emx)

+

Ex j/wu (-jB Emx - dx Emz) - Ey j/wu (dy Emz + jB Emy) +B/wu (Ex Emx + Ey Emy) - j/wu (Ex dx Emz + Ey dy Emz)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ E + + cfdfield_t + +
+

E-field of the mode

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber of the mode

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ axis + + int + +
+

Propagation axis (0=x, 1=y, 2=z)

+
+ 		+ required +
+ polarity + + int + +
+

Propagation direction (+1 for +ve, -1 for -ve)

+
+ 		+ required +
+ slices + + Sequence[slice] + +
+

epsilon[tuple(slices)] is used to select the portion of the grid to use + as the waveguide cross-section. slices[axis] should select only one item.

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ cfdfield_t + +
+

overlap_e normalized so that numpy.sum(overlap_e * E.conj()) == 1

+
+
+ cfdfield_t + +
+

over the retained overlap window.

+
+
+ + +
+ +
+ +
+ + +

+ expand_e + + +

+
expand_e(
+    E: cfdfield,
+    wavenumber: complex,
+    dxes: dx_lists_t,
+    axis: int,
+    polarity: int,
+    slices: Sequence[slice],
+) -> cfdfield_t
+
+ +
+ +

Given an eigenmode obtained by solve_mode, expands the E-field from the 2D +slice where the mode was calculated to the entire domain (along the propagation +axis). This assumes the epsilon cross-section remains constant throughout the +entire domain; it is up to the caller to truncate the expansion to any regions +where it is valid.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ E + + cfdfield + +
+

E-field of the mode

+
+ 		+ required +
+ wavenumber + + complex + +
+

Wavenumber of the mode

+
+ 		+ required +
+ dxes + + dx_lists_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types

+
+ 		+ required +
+ axis + + int + +
+

Propagation axis (0=x, 1=y, 2=z)

+
+ 		+ required +
+ polarity + + int + +
+

Propagation direction (+1 for +ve, -1 for -ve)

+
+ 		+ required +
+ slices + + Sequence[slice] + +
+

epsilon[tuple(slices)] is used to select the portion of the grid to use + as the waveguide cross-section. slices[axis] should select only one item.

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ cfdfield_t + +
+

E, with the original field expanded along the specified axis.

+
+
+ + +
+ Notes +

This helper assumes that the waveguide cross-section remains constant +along the propagation axis and applies the phase factor

+
\[ +e^{-i \, \mathrm{polarity} \, wavenumber \, \Delta z} +\]
+

to each copied slice.

+
+ +
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ meanas.fdfd.waveguide_cyl + + +

+ +
+ +

Operators and helper functions for cylindrical waveguides with unchanging cross-section.

+

Waveguide operator is derived according to 10.1364/OL.33.001848.

+

As in waveguide_2d, the propagation dependence is separated from the +transverse solve. Here the propagation coordinate is the bend angle \theta, +and the fields are assumed to have the form

+
\[ +\vec{E}(r, y, \theta), \vec{H}(r, y, \theta) \propto e^{-\imath m \theta}, +\]
+

where m is the angular wavenumber returned by solve_mode(s). It is often +convenient to introduce the corresponding linear wavenumber

+
\[ +\beta = \frac{m}{r_{\min}}, +\]
+

so that the cylindrical problem resembles the straight-waveguide problem with +additional metric factors.

+

Those metric factors live on the staggered radial Yee grids. If the left edge of +the computational window is at r = r_{\min}, define the electric-grid and +magnetic-grid radial sample locations by

+
\[ +\begin{aligned} +r_a(n) &= r_{\min} + \sum_{j \le n} \Delta r_{e, j}, \\ +r_b\!\left(n + \tfrac{1}{2}\right) &= r_{\min} + \tfrac{1}{2}\Delta r_{e, n} + + \sum_{j < n} \Delta r_{h, j}, +\end{aligned} +\]
+

and from them the diagonal metric matrices

+
\[ +\begin{aligned} +T_a &= \operatorname{diag}(r_a / r_{\min}), \\ +T_b &= \operatorname{diag}(r_b / r_{\min}). +\end{aligned} +\]
+

With the same forward/backward derivative notation used in waveguide_2d, the +coordinate-transformed discrete curl equations used here are

+
\[ +\begin{aligned} +-\imath \omega \mu_{rr} H_r &= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\ +-\imath \omega \mu_{yy} H_y &= -\imath \beta T_b^{-1} E_r + - T_b^{-1} \tilde{\partial}_r (T_a E_z), \\ +-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r, \\ +\imath \beta H_y &= -\imath \omega T_b \epsilon_{rr} E_r - T_b \hat{\partial}_y H_z, \\ +\imath \beta H_r &= \imath \omega T_a \epsilon_{yy} E_y + - T_b T_a^{-1} \hat{\partial}_r (T_b H_z), \\ +\imath \omega E_z &= T_a \epsilon_{zz}^{-1} + \left(\hat{\partial}_r H_y - \hat{\partial}_y H_r\right). +\end{aligned} +\]
+

The first three equations are the cylindrical analogue of the straight-guide +relations for H_r, H_y, and H_z. The next two are the metric-weighted +versions of the straight-guide identities for \imath \beta H_y and +\imath \beta H_r, and the last equation plays the same role as the +longitudinal E_z reconstruction in waveguide_2d.

+

Following the same elimination steps as in waveguide_2d, apply +\imath \beta \tilde{\partial}_r and \imath \beta \tilde{\partial}_y to the +equation for E_z, substitute for \imath \beta H_r and \imath \beta H_y, +and then eliminate H_z with

+
\[ +H_z = \frac{1}{-\imath \omega \mu_{zz}} +\left(\tilde{\partial}_r E_y - \tilde{\partial}_y E_r\right). +\]
+

This yields the transverse electric eigenproblem implemented by +cylindrical_operator(...):

+
\[ +\beta^2 +\begin{bmatrix} E_r \\ E_y \end{bmatrix} += +\left( +\omega^2 +\begin{bmatrix} +T_b^2 \mu_{yy} \epsilon_{xx} & 0 \\ +0 & T_a^2 \mu_{xx} \epsilon_{yy} +\end{bmatrix} ++ +\begin{bmatrix} +-T_b \mu_{yy} \hat{\partial}_y \\ + T_a \mu_{xx} \hat{\partial}_x +\end{bmatrix} +T_b \mu_{zz}^{-1} +\begin{bmatrix} +-\tilde{\partial}_y & \tilde{\partial}_x +\end{bmatrix} ++ +\begin{bmatrix} +\tilde{\partial}_x \\ +\tilde{\partial}_y +\end{bmatrix} +T_a \epsilon_{zz}^{-1} +\begin{bmatrix} +\hat{\partial}_x T_b \epsilon_{xx} & +\hat{\partial}_y T_a \epsilon_{yy} +\end{bmatrix} +\right) +\begin{bmatrix} E_r \\ E_y \end{bmatrix}. +\]
+

Since \beta = m / r_{\min}, the solver implemented in this file returns the +angular wavenumber m, while the operator itself is most naturally written in +terms of the linear quantity \beta. The helpers below reconstruct the full +field components from the solved transverse eigenvector and then normalize the +mode to unit forward power with the same discrete longitudinal Poynting inner +product used by waveguide_2d.

+

As in the straight-waveguide case, all functions here assume a 2D grid:

+

dxes = [[[dr_e_0, dr_e_1, ...], [dy_e_0, ...]], [[dr_h_0, ...], [dy_h_0, ...]]].

+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ cylindrical_operator + + +

+
cylindrical_operator(
+    omega: float,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    rmin: float,
+) -> sparse.sparray
+
+ +
+ +

Cylindrical coordinate waveguide operator of the form

+
\[ + (\omega^2 \begin{bmatrix} T_b T_b \mu_{yy} \epsilon_{xx} & 0 \\ + 0 & T_a T_a \mu_{xx} \epsilon_{yy} \end{bmatrix} + + \begin{bmatrix} -T_b \mu_{yy} \hat{\partial}_y \\ + T_a \mu_{xx} \hat{\partial}_x \end{bmatrix} T_b \mu_{zz}^{-1} + \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} + + \begin{bmatrix} \tilde{\partial}_x \\ + \tilde{\partial}_y \end{bmatrix} T_a \epsilon_{zz}^{-1} + \begin{bmatrix} \hat{\partial}_x T_b \epsilon_{xx} & \hat{\partial}_y T_a \epsilon_{yy} \end{bmatrix}) + \begin{bmatrix} E_r \\ + E_y \end{bmatrix} +\]
+

for use with a field vector of the form [E_r, E_y].

+

This operator can be used to form an eigenvalue problem of the form + A @ [E_r, E_y] = beta**2 * [E_r, E_y]

+

which can then be solved for the eigenmodes of the system +(an exp(-i * angular_wavenumber * theta) theta-dependence is assumed for +the fields, with beta = angular_wavenumber / rmin).

+

(NOTE: See module docs and 10.1364/OL.33.001848)

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ omega + + float + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator

+
+
+ + +
+ +
+ +
+ + +

+ solve_modes + + +

+
solve_modes(
+    mode_numbers: Sequence[int],
+    omega: float,
+    dxes: dx_lists2_t,
+    epsilon: vfdslice,
+    rmin: float,
+    mode_margin: int = 2,
+) -> tuple[
+    NDArray[numpy.complex128], NDArray[numpy.complex128]
+]
+
+ +
+ +

Given a 2d (r, y) slice of epsilon, attempts to solve for the eigenmode + of the bent waveguide with the specified mode number.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ mode_number + + +
+

Number of the mode, 0-indexed

+
+ 		+ required +
+ omega + + float + +
+

Angular frequency of the simulation

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types. + The first coordinate is assumed to be r, the second is y.

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Dielectric constant

+
+ 		+ required +
+ rmin + + float + +
+

Radius of curvature for the simulation. This should be the minimum value of + r within the simulation domain.

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
Name TypeDescription
e_xys + NDArray[complex128] + +
+

NDArray of vfdfield_t specifying fields. First dimension is mode number.

+
+
angular_wavenumbers + NDArray[complex128] + +
+

list of wavenumbers in 1/rad units.

+
+
+ + +
+ +
+ +
+ + +

+ solve_mode + + +

+
solve_mode(
+    mode_number: int, *args: Any, **kwargs: Any
+) -> tuple[vcfdslice, complex]
+
+ +
+ +

Wrapper around solve_modes() that solves for a single mode.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ mode_number + + int + +
+

0-indexed mode number to solve for

+
+ 		+ required +
+ *args + + Any + +
+

passed to solve_modes()

+
+ 		+ () +
+ **kwargs + + Any + +
+

passed to solve_modes()

+
+ 		+ {} +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[vcfdslice, complex] + +
+

(e_xy, angular_wavenumber)

+
+
+ + +
+ +
+ +
+ + +

+ linear_wavenumbers + + +

+
linear_wavenumbers(
+    e_xys: list[vcfdfield2_t],
+    angular_wavenumbers: ArrayLike,
+    epsilon: vfdslice,
+    dxes: dx_lists2_t,
+    rmin: float,
+) -> NDArray[numpy.complex128]
+
+ +
+ +

Calculate linear wavenumbers (1/distance) based on angular wavenumbers (1/rad) + and the mode's energy distribution.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e_xys + + list[vcfdfield2_t] + +
+

Vectorized mode fields with shape (num_modes, 2 * x *y)

+
+ 		+ required +
+ angular_wavenumbers + + ArrayLike + +
+

Wavenumbers assuming fields have theta-dependence of +exp(-i * angular_wavenumber * theta). They should satisfy +operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid with shape (3, x, y)

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ NDArray[complex128] + +
+

NDArray containing the calculated linear (1/distance) wavenumbers

+
+
+ + +
+ +
+ +
+ + +

+ exy2h + + +

+
exy2h(
+    angular_wavenumber: complex,
+    omega: float,
+    dxes: dx_lists2_t,
+    rmin: float,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Operator which transforms the vector e_xy containing the vectorized E_r and E_y fields, + into a vectorized H containing all three H components

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ angular_wavenumber + + complex + +
+

Wavenumber assuming fields have theta-dependence of +exp(-i * angular_wavenumber * theta). It should satisfy +operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy

+
+ 		+ required +
+ omega + + float + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representing the operator.

+
+
+ + +
+ +
+ +
+ + +

+ exy2e + + +

+
exy2e(
+    angular_wavenumber: complex,
+    omega: float,
+    dxes: dx_lists2_t,
+    rmin: float,
+    epsilon: vfdslice,
+) -> sparse.sparray
+
+ +
+ +

Operator which transforms the vector e_xy containing the vectorized E_r and E_y fields, + into a vectorized E containing all three E components

+

Unlike the straight waveguide case, the H_z components do not cancel and must be calculated +from E_r and E_y in order to then calculate E_z.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ angular_wavenumber + + complex + +
+

Wavenumber assuming fields have theta-dependence of +exp(-i * angular_wavenumber * theta). It should satisfy +operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy

+
+ 		+ required +
+ omega + + float + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representing the operator.

+
+
+ + +
+ +
+ +
+ + +

+ e2h + + +

+
e2h(
+    angular_wavenumber: complex,
+    omega: float,
+    dxes: dx_lists2_t,
+    rmin: float,
+    mu: vfdslice | None = None,
+) -> sparse.sparray
+
+ +
+ +

Returns an operator which, when applied to a vectorized E eigenfield, produces + the vectorized H eigenfield.

+

This operator is created directly from the initial coordinate-transformed equations: +$$ +\begin{aligned} +-\imath \omega \mu_{rr} H_r &= \tilde{\partial}y E_z + \imath \beta T_a^{-1} E_y, \ +-\imath \omega \mu E_r + - T_b^{-1} \tilde{\partial}} H_y &= -\imath \beta T_b^{-1r (T_a E_z), \ +-\imath \omega \mu_y E_r, +\end{aligned} +$$} H_z &= \tilde{\partial}_r E_y - \tilde{\partial

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ angular_wavenumber + + complex + +
+

Wavenumber assuming fields have theta-dependence of +exp(-i * angular_wavenumber * theta). It should satisfy +operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy

+
+ 		+ required +
+ omega + + float + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ sparray + +
+

Sparse matrix representation of the operator.

+
+
+ + +
+ +
+ +
+ + +

+ dxes2T + + +

+
dxes2T(
+    dxes: dx_lists2_t, rmin: float
+) -> tuple[NDArray[numpy.float64], NDArray[numpy.float64]]
+
+ +
+ +

Construct the cylindrical metric matrices \(T_a\) and \(T_b\).

+

T_a is sampled on the E-grid radial locations, while T_b is sampled on +the staggered H-grid radial locations. These are the diagonal matrices that +convert the straight-waveguide algebra into its cylindrical counterpart.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ + +

Returns:

+ + + + + + + + + + + + + +
TypeDescription
+ tuple[NDArray[float64], NDArray[float64]] + +
+

Sparse diagonal matrices (T_a, T_b).

+
+
+ + +
+ +
+ +
+ + +

+ normalized_fields_e + + +

+
normalized_fields_e(
+    e_xy: vcfdfield2,
+    angular_wavenumber: complex,
+    omega: float,
+    dxes: dx_lists2_t,
+    rmin: float,
+    epsilon: vfdslice,
+    mu: vfdslice | None = None,
+    prop_phase: float = 0,
+) -> tuple[vcfdslice_t, vcfdslice_t]
+
+ +
+ +

Given a vector e_xy containing the vectorized E_r and E_y fields, +returns normalized, vectorized E and H fields for the system.

+ + +

Parameters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeDescriptionDefault
+ e_xy + + vcfdfield2 + +
+

Vector containing E_r and E_y fields

+
+ 		+ required +
+ angular_wavenumber + + complex + +
+

Wavenumber assuming fields have theta-dependence of +exp(-i * angular_wavenumber * theta). It should satisfy +operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy

+
+ 		+ required +
+ omega + + float + +
+

The angular frequency of the system

+
+ 		+ required +
+ dxes + + dx_lists2_t + +
+

Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types (2D)

+
+ 		+ required +
+ rmin + + float + +
+

Radius at the left edge of the simulation domain (at minimum 'x')

+
+ 		+ required +
+ epsilon + + vfdslice + +
+

Vectorized dielectric constant grid

+
+ 		+ required +
+ mu + + vfdslice | None + +
+

Vectorized magnetic permeability grid (default 1 everywhere)

+
+ 		+ None +
+ prop_phase + + float + +
+

Phase shift (dz * corrected_wavenumber) over 1 cell in propagation direction. + Default 0 (continuous propagation direction, i.e. dz->0).

+
+ 		+ 0 +
+ + +

Returns:

+ + + + + + + + + + + + + + + + + +
TypeDescription
+ vcfdslice_t + +
+

(e, h), where each field is vectorized, normalized,

+
+
+ vcfdslice_t + +
+

and contains all three vector components.

+
+
+ + +
+ Notes +

The normalization step is delegated to _normalized_fields(...), which +enforces unit forward power under the discrete inner product

+
\[ +\frac{1}{2}\int (E_r H_y^* - E_y H_r^*) \, dr \, dy. +\]
+

The angular wavenumber m is first converted into the full three-component +fields, then the overall complex phase and sign are fixed so the result is +reproducible for symmetric modes.

+
+ +
+ +
+ + + +
+ +
+ +
+ + + + + + + + + + + + + +
+
+ + + +
+ + + +
+ + + +
+
+
+
+ + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/assets/_mkdocstrings.css b/assets/_mkdocstrings.css new file mode 100644 index 0000000..854048c --- /dev/null +++ b/assets/_mkdocstrings.css @@ -0,0 +1,237 @@ + +/* Avoid breaking parameter names, etc. in table cells. */ +.doc-contents td code { + word-break: normal !important; +} + +/* No line break before first paragraph of descriptions. */ +.doc-md-description, +.doc-md-description>p:first-child { + display: inline; +} + +/* No text transformation from Material for MkDocs for H5 headings. */ +.md-typeset h5 .doc-object-name { + text-transform: none; +} + +/* Max width for docstring sections tables. */ +.doc .md-typeset__table, +.doc .md-typeset__table table { + display: table !important; + width: 100%; +} + +.doc .md-typeset__table tr { + display: table-row; +} + +/* Defaults in Spacy table style. */ +.doc-param-default, +.doc-type_param-default { + float: right; +} + +/* Parameter headings must be inline, not blocks. */ +.doc-heading-parameter, +.doc-heading-type_parameter { + display: inline; +} + +/* Default font size for parameter headings. */ +.md-typeset .doc-heading-parameter { + font-size: inherit; +} + +/* Prefer space on the right, not the left of parameter permalinks. */ +.doc-heading-parameter .headerlink, +.doc-heading-type_parameter .headerlink { + margin-left: 0 !important; + margin-right: 0.2rem; +} + +/* Backward-compatibility: docstring section titles in bold. */ +.doc-section-title { + font-weight: bold; +} + +/* Backlinks crumb separator. */ +.doc-backlink-crumb { + display: inline-flex; + gap: .2rem; + white-space: nowrap; + align-items: center; + vertical-align: middle; +} +.doc-backlink-crumb:not(:first-child)::before { + background-color: var(--md-default-fg-color--lighter); + content: ""; + display: inline; + height: 1rem; + --md-path-icon: url('data:image/svg+xml;charset=utf-8,'); + -webkit-mask-image: var(--md-path-icon); + mask-image: var(--md-path-icon); + width: 1rem; +} +.doc-backlink-crumb.last { + font-weight: bold; +} + +/* Symbols in Navigation and ToC. */ +:root, :host, +[data-md-color-scheme="default"] { + --doc-symbol-parameter-fg-color: #df50af; + --doc-symbol-type_parameter-fg-color: #df50af; + --doc-symbol-attribute-fg-color: #953800; + --doc-symbol-function-fg-color: #8250df; + --doc-symbol-method-fg-color: #8250df; + --doc-symbol-class-fg-color: #0550ae; + --doc-symbol-type_alias-fg-color: #0550ae; + --doc-symbol-module-fg-color: #5cad0f; + + --doc-symbol-parameter-bg-color: #df50af1a; + --doc-symbol-type_parameter-bg-color: #df50af1a; + --doc-symbol-attribute-bg-color: #9538001a; + --doc-symbol-function-bg-color: #8250df1a; + --doc-symbol-method-bg-color: #8250df1a; + --doc-symbol-class-bg-color: #0550ae1a; + --doc-symbol-type_alias-bg-color: #0550ae1a; + --doc-symbol-module-bg-color: #5cad0f1a; +} + +[data-md-color-scheme="slate"] { + --doc-symbol-parameter-fg-color: #ffa8cc; + 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"param"; +} + +code.doc-symbol-type_parameter, +a code.doc-symbol-type_parameter { + color: var(--doc-symbol-type_parameter-fg-color); + background-color: var(--doc-symbol-type_parameter-bg-color); +} + +code.doc-symbol-type_parameter::after { + content: "type-param"; +} + +code.doc-symbol-attribute, +a code.doc-symbol-attribute { + color: var(--doc-symbol-attribute-fg-color); + background-color: var(--doc-symbol-attribute-bg-color); +} + +code.doc-symbol-attribute::after { + content: "attr"; +} + +code.doc-symbol-function, +a code.doc-symbol-function { + color: var(--doc-symbol-function-fg-color); + background-color: var(--doc-symbol-function-bg-color); +} + +code.doc-symbol-function::after { + content: "func"; +} + +code.doc-symbol-method, +a code.doc-symbol-method { + color: var(--doc-symbol-method-fg-color); + background-color: var(--doc-symbol-method-bg-color); +} + +code.doc-symbol-method::after { + content: "meth"; +} + +code.doc-symbol-class, +a code.doc-symbol-class { + color: var(--doc-symbol-class-fg-color); + background-color: var(--doc-symbol-class-bg-color); +} + +code.doc-symbol-class::after { + content: "class"; +} + + +code.doc-symbol-type_alias, +a code.doc-symbol-type_alias { + color: var(--doc-symbol-type_alias-fg-color); + background-color: var(--doc-symbol-type_alias-bg-color); +} + +code.doc-symbol-type_alias::after { + content: "type"; +} + +code.doc-symbol-module, +a code.doc-symbol-module { + color: var(--doc-symbol-module-fg-color); + background-color: var(--doc-symbol-module-bg-color); +} + +code.doc-symbol-module::after { + content: "mod"; +} + +.doc-signature .autorefs { + color: inherit; + border-bottom: 1px dotted currentcolor; +} + +/* Source code blocks (admonitions). */ +:root { + --md-admonition-icon--mkdocstrings-source: url('data:image/svg+xml;charset=utf-8,') +} +.md-typeset .admonition.mkdocstrings-source, +.md-typeset details.mkdocstrings-source { + border: none; + padding: 0; +} +.md-typeset 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foreignObject{line-height:normal;overflow:visible}.label div .edgeLabel{color:var(--md-mermaid-label-fg-color)}.edgeLabel,.edgeLabel p,.label div .edgeLabel{background-color:var(--md-mermaid-label-bg-color)}.edgeLabel,.edgeLabel p{fill:var(--md-mermaid-label-bg-color);color:var(--md-mermaid-edge-color)}.edgePath .path,.flowchart-link{stroke:var(--md-mermaid-edge-color)}.edgePath .arrowheadPath{fill:var(--md-mermaid-edge-color);stroke:none}.cluster rect{fill:var(--md-default-fg-color--lightest);stroke:var(--md-default-fg-color--lighter)}.cluster span{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}g #flowchart-circleEnd,g #flowchart-circleStart,g #flowchart-crossEnd,g #flowchart-crossStart,g #flowchart-pointEnd,g #flowchart-pointStart{stroke:none}.classDiagramTitleText{fill:var(--md-mermaid-label-fg-color)}g.classGroup line,g.classGroup rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}g.classGroup text{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.classLabel .box{fill:var(--md-mermaid-label-bg-color);background-color:var(--md-mermaid-label-bg-color);opacity:1}.classLabel .label{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.node .divider{stroke:var(--md-mermaid-node-fg-color)}.relation{stroke:var(--md-mermaid-edge-color)}.cardinality{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.cardinality text{fill:inherit!important}defs marker.marker.composition.class path,defs marker.marker.dependency.class path,defs marker.marker.extension.class path{fill:var(--md-mermaid-edge-color)!important;stroke:var(--md-mermaid-edge-color)!important}defs marker.marker.aggregation.class path{fill:var(--md-mermaid-label-bg-color)!important;stroke:var(--md-mermaid-edge-color)!important}.statediagramTitleText{fill:var(--md-mermaid-label-fg-color)}g.stateGroup rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}g.stateGroup .state-title{fill:var(--md-mermaid-label-fg-color)!important;font-family:var(--md-mermaid-font-family)}g.stateGroup .composit{fill:var(--md-mermaid-label-bg-color)}.nodeLabel,.nodeLabel p{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}a .nodeLabel{text-decoration:underline}.node circle.state-end,.node circle.state-start,.start-state{fill:var(--md-mermaid-edge-color);stroke:none}.end-state-inner,.end-state-outer{fill:var(--md-mermaid-edge-color)}.end-state-inner,.node circle.state-end{stroke:var(--md-mermaid-label-bg-color)}.transition{stroke:var(--md-mermaid-edge-color)}[id^=state-fork] rect,[id^=state-join] rect{fill:var(--md-mermaid-edge-color)!important;stroke:none!important}.statediagram-cluster.statediagram-cluster .inner{fill:var(--md-default-bg-color)}.statediagram-cluster rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}.statediagram-state rect.divider{fill:var(--md-default-fg-color--lightest);stroke:var(--md-default-fg-color--lighter)}defs #statediagram-barbEnd{stroke:var(--md-mermaid-edge-color)}[id^=entity] path,[id^=entity] rect{fill:var(--md-default-bg-color)}.relationshipLine{stroke:var(--md-mermaid-edge-color)}defs .marker.oneOrMore.er *,defs .marker.onlyOne.er *,defs .marker.zeroOrMore.er *,defs .marker.zeroOrOne.er *{stroke:var(--md-mermaid-edge-color)!important}text:not([class]):last-child{fill:var(--md-mermaid-label-fg-color)}.actor{fill:var(--md-mermaid-sequence-actor-bg-color);stroke:var(--md-mermaid-sequence-actor-border-color)}text.actor>tspan{fill:var(--md-mermaid-sequence-actor-fg-color);font-family:var(--md-mermaid-font-family)}line{stroke:var(--md-mermaid-sequence-actor-line-color)}.actor-man circle,.actor-man line{fill:var(--md-mermaid-sequence-actorman-bg-color);stroke:var(--md-mermaid-sequence-actorman-line-color)}.messageLine0,.messageLine1{stroke:var(--md-mermaid-sequence-message-line-color)}.note{fill:var(--md-mermaid-sequence-note-bg-color);stroke:var(--md-mermaid-sequence-note-border-color)}.loopText,.loopText>tspan,.messageText,.noteText>tspan{stroke:none;font-family:var(--md-mermaid-font-family)!important}.messageText{fill:var(--md-mermaid-sequence-message-fg-color)}.loopText,.loopText>tspan{fill:var(--md-mermaid-sequence-loop-fg-color)}.noteText>tspan{fill:var(--md-mermaid-sequence-note-fg-color)}#arrowhead path{fill:var(--md-mermaid-sequence-message-line-color);stroke:none}.loopLine{fill:var(--md-mermaid-sequence-loop-bg-color);stroke:var(--md-mermaid-sequence-loop-border-color)}.labelBox{fill:var(--md-mermaid-sequence-label-bg-color);stroke:none}.labelText,.labelText>span{fill:var(--md-mermaid-sequence-label-fg-color);font-family:var(--md-mermaid-font-family)}.sequenceNumber{fill:var(--md-mermaid-sequence-number-fg-color)}rect.rect{fill:var(--md-mermaid-sequence-box-bg-color);stroke:none}rect.rect+text.text{fill:var(--md-mermaid-sequence-box-fg-color)}defs #sequencenumber{fill:var(--md-mermaid-sequence-number-bg-color)!important}";var so,is=0;function as(){return typeof mermaid=="undefined"||mermaid instanceof Element?_t("https://unpkg.com/mermaid@11/dist/mermaid.min.js"):$(void 0)}function Zn(e){return e.classList.remove("mermaid"),so||(so=as().pipe(O(()=>mermaid.initialize({startOnLoad:!1,themeCSS:Xn,sequence:{actorFontSize:"16px",messageFontSize:"16px",noteFontSize:"16px"}})),m(()=>{}),Z(1))),so.subscribe(()=>go(null,null,function*(){e.classList.add("mermaid");let t=`__mermaid_${is++}`,r=x("div",{class:"mermaid"}),o=e.textContent,{svg:n,fn:i}=yield mermaid.render(t,o),s=r.attachShadow({mode:"closed"});s.innerHTML=n,e.replaceWith(r),i==null||i(s)})),so.pipe(m(()=>({ref:e})))}var ei=x("table");function ti(e){return e.replaceWith(ei),ei.replaceWith(Un(e)),$({ref:e})}function ss(e){let t=e.find(r=>r.checked)||e[0];return L(...e.map(r=>h(r,"change").pipe(m(()=>j(`label[for="${r.id}"]`))))).pipe(Q(j(`label[for="${t.id}"]`)),m(r=>({active:r})))}function ri(e,{viewport$:t,target$:r}){let o=j(".tabbed-labels",e),n=M(":scope > input",e),i=no("prev");e.append(i);let s=no("next");return e.append(s),H(()=>{let a=new T,c=a.pipe(oe(),ae(!0));z([a,Le(e),mt(e)]).pipe(W(c),$e(1,ye)).subscribe({next([{active:p},l]){let f=Be(p),{width:u}=de(p);e.style.setProperty("--md-indicator-x",`${f.x}px`),e.style.setProperty("--md-indicator-width",`${u}px`);let d=gr(o);(f.xd.x+l.width)&&o.scrollTo({left:Math.max(0,f.x-16),behavior:"smooth"})},complete(){e.style.removeProperty("--md-indicator-x"),e.style.removeProperty("--md-indicator-width")}}),z([Ge(o),Le(o)]).pipe(W(c)).subscribe(([p,l])=>{let f=At(o);i.hidden=p.x<16,s.hidden=p.x>f.width-l.width-16}),L(h(i,"click").pipe(m(()=>-1)),h(s,"click").pipe(m(()=>1))).pipe(W(c)).subscribe(p=>{let{width:l}=de(o);o.scrollBy({left:l*p,behavior:"smooth"})}),r.pipe(W(c),g(p=>n.includes(p))).subscribe(p=>p.click()),o.classList.add("tabbed-labels--linked");for(let p of n){let l=j(`label[for="${p.id}"]`);l.replaceChildren(x("a",{href:`#${l.htmlFor}`,tabIndex:-1},...Array.from(l.childNodes))),h(l.firstElementChild,"click").pipe(W(c),g(f=>!(f.metaKey||f.ctrlKey)),O(f=>{f.preventDefault(),f.stopPropagation()})).subscribe(()=>{history.replaceState({},"",`#${l.htmlFor}`),l.click()})}return V("content.tabs.link")&&a.pipe(Ie(1),te(t)).subscribe(([{active:p},{offset:l}])=>{let f=p.innerText.trim();if(p.hasAttribute("data-md-switching"))p.removeAttribute("data-md-switching");else{let u=e.offsetTop-l.y;for(let v of M("[data-tabs]"))for(let S of M(":scope > input",v)){let X=j(`label[for="${S.id}"]`);if(X!==p&&X.innerText.trim()===f){X.setAttribute("data-md-switching",""),S.click();break}}window.scrollTo({top:e.offsetTop-u});let d=__md_get("__tabs")||[];__md_set("__tabs",[...new Set([f,...d])])}}),a.pipe(W(c)).subscribe(()=>{for(let p of M("audio, video",e))p.offsetWidth&&p.autoplay?p.play().catch(()=>{}):p.pause()}),ss(n).pipe(O(p=>a.next(p)),A(()=>a.complete()),m(p=>P({ref:e},p)))}).pipe(et(pe))}function oi(e,t){let{viewport$:r,target$:o,print$:n}=t;return L(...M(".annotate:not(.highlight)",e).map(i=>zn(i,{target$:o,print$:n})),...M("pre:not(.mermaid) > code",e).map(i=>Yn(i,{target$:o,print$:n})),...M("a",e).map(i=>Jn(i,t)),...M("pre.mermaid",e).map(i=>Zn(i)),...M("table:not([class])",e).map(i=>ti(i)),...M("details",e).map(i=>Bn(i,{target$:o,print$:n})),...M("[data-tabs]",e).map(i=>ri(i,{viewport$:r,target$:o})),...M("[title]:not([data-preview])",e).filter(()=>V("content.tooltips")).map(i=>Xe(i,{viewport$:r})),...M(".footnote-ref",e).filter(()=>V("content.footnote.tooltips")).map(i=>Vt(i,{content$:new F(s=>{let a=new URL(i.href).hash.slice(1),c=Array.from(document.getElementById(a).cloneNode(!0).children),p=wr(...c);return s.next(p),document.body.append(p),()=>p.remove()}),viewport$:r})))}function cs(e,{alert$:t}){return t.pipe(b(r=>L($(!0),$(!1).pipe(nt(2e3))).pipe(m(o=>({message:r,active:o})))))}function ni(e,t){let r=j(".md-typeset",e);return H(()=>{let o=new T;return o.subscribe(({message:n,active:i})=>{e.classList.toggle("md-dialog--active",i),r.textContent=n}),cs(e,t).pipe(O(n=>o.next(n)),A(()=>o.complete()),m(n=>P({ref:e},n)))})}var ps=0;function ls(e,t){document.body.append(e);let{width:r}=de(e);e.style.setProperty("--md-tooltip-width",`${r}px`),e.remove();let o=vr(t),n=typeof o!="undefined"?Ge(o):$({x:0,y:0}),i=L(Ye(t),it(t)).pipe(Y());return z([i,n]).pipe(m(([s,a])=>{let{x:c,y:p}=Be(t),l=de(t),f=t.closest("table");return f&&t.parentElement&&(c+=f.offsetLeft+t.parentElement.offsetLeft,p+=f.offsetTop+t.parentElement.offsetTop),{active:s,offset:{x:c-a.x+l.width/2-r/2,y:p-a.y+l.height+8}}}))}function ii(e){let t=e.title;if(!t.length)return y;let r=`__tooltip_${ps++}`,o=Dt(r,"inline"),n=j(".md-typeset",o);return n.innerHTML=t,H(()=>{let i=new T;return i.subscribe({next({offset:s}){o.style.setProperty("--md-tooltip-x",`${s.x}px`),o.style.setProperty("--md-tooltip-y",`${s.y}px`)},complete(){o.style.removeProperty("--md-tooltip-x"),o.style.removeProperty("--md-tooltip-y")}}),L(i.pipe(g(({active:s})=>s)),i.pipe(Ae(250),g(({active:s})=>!s))).subscribe({next({active:s}){s?(e.insertAdjacentElement("afterend",o),e.setAttribute("aria-describedby",r),e.removeAttribute("title")):(o.remove(),e.removeAttribute("aria-describedby"),e.setAttribute("title",t))},complete(){o.remove(),e.removeAttribute("aria-describedby"),e.setAttribute("title",t)}}),i.pipe($e(16,ye)).subscribe(({active:s})=>{o.classList.toggle("md-tooltip--active",s)}),i.pipe(gt(125,ye),g(()=>!!e.offsetParent),m(()=>e.offsetParent.getBoundingClientRect()),m(({x:s})=>s)).subscribe({next(s){s?o.style.setProperty("--md-tooltip-0",`${-s}px`):o.style.removeProperty("--md-tooltip-0")},complete(){o.style.removeProperty("--md-tooltip-0")}}),ls(o,e).pipe(O(s=>i.next(s)),A(()=>i.complete()),m(s=>P({ref:e},s)))}).pipe(et(pe))}function ms({viewport$:e}){if(!V("header.autohide"))return $(!1);let t=e.pipe(m(({offset:{y:n}})=>n),ot(2,1),m(([n,i])=>[nMath.abs(i-n.y)>100),m(([,[n]])=>n),Y()),o=Je("search");return z([e,o]).pipe(m(([{offset:n},i])=>n.y>400&&!i),Y(),b(n=>n?r:$(!1)),Q(!1))}function ai(e,t){return H(()=>z([Le(e),ms(t)])).pipe(m(([{height:r},o])=>({height:r,hidden:o})),Y((r,o)=>r.height===o.height&&r.hidden===o.hidden),Z(1))}function si(e,{header$:t,main$:r}){return H(()=>{let o=new T,n=o.pipe(oe(),ae(!0));o.pipe(ne("active"),Pe(t)).subscribe(([{active:s},{hidden:a}])=>{e.classList.toggle("md-header--shadow",s&&!a),e.hidden=a});let i=fe(M("[title]",e)).pipe(g(()=>V("content.tooltips")),J(s=>ii(s)));return r.subscribe(o),t.pipe(W(n),m(s=>P({ref:e},s)),Ve(i.pipe(W(n))))})}function fs(e,{viewport$:t,header$:r}){return Er(e,{viewport$:t,header$:r}).pipe(m(({offset:{y:o}})=>{let{height:n}=de(e);return{active:n>0&&o>=n}}),ne("active"))}function ci(e,t){return H(()=>{let r=new T;r.subscribe({next({active:n}){e.classList.toggle("md-header__title--active",n)},complete(){e.classList.remove("md-header__title--active")}});let o=ue(".md-content h1");return typeof o=="undefined"?y:fs(o,t).pipe(O(n=>r.next(n)),A(()=>r.complete()),m(n=>P({ref:e},n)))})}function pi(e,{viewport$:t,header$:r}){let o=r.pipe(m(({height:i})=>i),Y()),n=o.pipe(b(()=>Le(e).pipe(m(({height:i})=>({top:e.offsetTop,bottom:e.offsetTop+i})),ne("bottom"))));return z([o,n,t]).pipe(m(([i,{top:s,bottom:a},{offset:{y:c},size:{height:p}}])=>(p=Math.max(0,p-Math.max(0,s-c,i)-Math.max(0,p+c-a)),{offset:s-i,height:p,active:s-i<=c})),Y((i,s)=>i.offset===s.offset&&i.height===s.height&&i.active===s.active))}function us(e){let t=__md_get("__palette")||{index:e.findIndex(o=>matchMedia(o.getAttribute("data-md-color-media")).matches)},r=Math.max(0,Math.min(t.index,e.length-1));return $(...e).pipe(J(o=>h(o,"change").pipe(m(()=>o))),Q(e[r]),m(o=>({index:e.indexOf(o),color:{media:o.getAttribute("data-md-color-media"),scheme:o.getAttribute("data-md-color-scheme"),primary:o.getAttribute("data-md-color-primary"),accent:o.getAttribute("data-md-color-accent")}})),Z(1))}function li(e){let t=M("input",e),r=x("meta",{name:"theme-color"});document.head.appendChild(r);let o=x("meta",{name:"color-scheme"});document.head.appendChild(o);let n=Wt("(prefers-color-scheme: light)");return H(()=>{let i=new T;return i.subscribe(s=>{if(document.body.setAttribute("data-md-color-switching",""),s.color.media==="(prefers-color-scheme)"){let a=matchMedia("(prefers-color-scheme: light)"),c=document.querySelector(a.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: 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fi(e,t){return e.protocol=t.protocol,e.hostname=t.hostname,e}function ds(e,t){let r=new Map;for(let o of M("url",e)){let n=j("loc",o),i=[fi(new URL(n.textContent),t)];r.set(`${i[0]}`,i);for(let s of M("[rel=alternate]",o)){let a=s.getAttribute("href");a!=null&&i.push(fi(new URL(a),t))}}return r}function kt(e){return En(new URL("sitemap.xml",e)).pipe(m(t=>ds(t,new URL(e))),ve(()=>$(new Map)),le())}function ui({document$:e}){let t=new Map;e.pipe(b(()=>M("link[rel=alternate]")),m(r=>new URL(r.href)),g(r=>!t.has(r.toString())),J(r=>kt(r).pipe(m(o=>[r,o]),ve(()=>y)))).subscribe(([r,o])=>{t.set(r.toString().replace(/\/$/,""),o)}),h(document.body,"click").pipe(g(r=>!r.metaKey&&!r.ctrlKey),b(r=>{if(r.target instanceof Element){let o=r.target.closest("a");if(o&&!o.target){let n=[...t].find(([f])=>o.href.startsWith(`${f}/`));if(typeof n=="undefined")return y;let[i,s]=n,a=we();if(a.href.startsWith(i))return y;let c=Te(),p=a.href.replace(c.base,"");p=`${i}/${p}`;let l=s.has(p.split("#")[0])?new 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ft=ai(Ce("header"),{viewport$:ke}),qt=ct.pipe(m(()=>Ce("main")),b(e=>pi(e,{viewport$:ke,header$:ft})),Z(1)),Hs=L(...me("consent").map(e=>An(e,{target$:Ht})),...me("dialog").map(e=>ni(e,{alert$:fo})),...me("palette").map(e=>li(e)),...me("progress").map(e=>mi(e,{progress$:uo})),...me("search").map(e=>_i(e,{index$:zi,keyboard$:mo})),...me("source").map(e=>$i(e))),$s=H(()=>L(...me("announce").map(e=>_n(e)),...me("content").map(e=>oi(e,{sitemap$:qi,viewport$:ke,target$:Ht,print$:Ni})),...me("content").map(e=>V("search.highlight")?Ai(e,{index$:zi,location$:Kt}):y),...me("header").map(e=>si(e,{viewport$:ke,header$:ft,main$:qt})),...me("header-title").map(e=>ci(e,{viewport$:ke,header$:ft})),...me("sidebar").map(e=>e.getAttribute("data-md-type")==="navigation"?eo(Vi,()=>lo(e,{viewport$:ke,header$:ft,main$:qt})):eo(Lr,()=>lo(e,{viewport$:ke,header$:ft,main$:qt}))),...me("tabs").map(e=>Pi(e,{viewport$:ke,header$:ft})),...me("toc").map(e=>Ri(e,{viewport$:ke,header$:ft,main$:qt,target$:Ht})),...me("top").map(e=>Ii(e,{viewport$:ke,header$:ft,main$:qt,target$:Ht})))),Ki=ct.pipe(b(()=>$s),Ve(Hs),Z(1));Ki.subscribe();window.document$=ct;window.location$=Kt;window.target$=Ht;window.keyboard$=mo;window.viewport$=ke;window.tablet$=Lr;window.screen$=Vi;window.print$=Ni;window.alert$=fo;window.progress$=uo;window.component$=Ki;})(); +//# sourceMappingURL=bundle.79ae519e.min.js.map + diff --git a/assets/javascripts/bundle.79ae519e.min.js.map b/assets/javascripts/bundle.79ae519e.min.js.map new file mode 100644 index 0000000..5cf0289 --- /dev/null +++ b/assets/javascripts/bundle.79ae519e.min.js.map @@ -0,0 +1,7 @@ +{ + "version": 3, + "sources": ["node_modules/focus-visible/dist/focus-visible.js", "node_modules/escape-html/index.js", "node_modules/clipboard/dist/clipboard.js", "src/templates/assets/javascripts/bundle.ts", "node_modules/tslib/tslib.es6.mjs", "node_modules/rxjs/src/internal/util/isFunction.ts", "node_modules/rxjs/src/internal/util/createErrorClass.ts", "node_modules/rxjs/src/internal/util/UnsubscriptionError.ts", "node_modules/rxjs/src/internal/util/arrRemove.ts", "node_modules/rxjs/src/internal/Subscription.ts", "node_modules/rxjs/src/internal/config.ts", "node_modules/rxjs/src/internal/scheduler/timeoutProvider.ts", "node_modules/rxjs/src/internal/util/reportUnhandledError.ts", "node_modules/rxjs/src/internal/util/noop.ts", "node_modules/rxjs/src/internal/NotificationFactories.ts", "node_modules/rxjs/src/internal/util/errorContext.ts", "node_modules/rxjs/src/internal/Subscriber.ts", "node_modules/rxjs/src/internal/symbol/observable.ts", "node_modules/rxjs/src/internal/util/identity.ts", "node_modules/rxjs/src/internal/util/pipe.ts", "node_modules/rxjs/src/internal/Observable.ts", "node_modules/rxjs/src/internal/util/lift.ts", "node_modules/rxjs/src/internal/operators/OperatorSubscriber.ts", "node_modules/rxjs/src/internal/scheduler/animationFrameProvider.ts", "node_modules/rxjs/src/internal/util/ObjectUnsubscribedError.ts", "node_modules/rxjs/src/internal/Subject.ts", "node_modules/rxjs/src/internal/BehaviorSubject.ts", "node_modules/rxjs/src/internal/scheduler/dateTimestampProvider.ts", "node_modules/rxjs/src/internal/ReplaySubject.ts", "node_modules/rxjs/src/internal/scheduler/Action.ts", "node_modules/rxjs/src/internal/scheduler/intervalProvider.ts", "node_modules/rxjs/src/internal/scheduler/AsyncAction.ts", "node_modules/rxjs/src/internal/Scheduler.ts", "node_modules/rxjs/src/internal/scheduler/AsyncScheduler.ts", "node_modules/rxjs/src/internal/scheduler/async.ts", "node_modules/rxjs/src/internal/scheduler/QueueAction.ts", "node_modules/rxjs/src/internal/scheduler/QueueScheduler.ts", "node_modules/rxjs/src/internal/scheduler/queue.ts", "node_modules/rxjs/src/internal/scheduler/AnimationFrameAction.ts", "node_modules/rxjs/src/internal/scheduler/AnimationFrameScheduler.ts", "node_modules/rxjs/src/internal/scheduler/animationFrame.ts", "node_modules/rxjs/src/internal/observable/empty.ts", "node_modules/rxjs/src/internal/util/isScheduler.ts", "node_modules/rxjs/src/internal/util/args.ts", "node_modules/rxjs/src/internal/util/isArrayLike.ts", "node_modules/rxjs/src/internal/util/isPromise.ts", "node_modules/rxjs/src/internal/util/isInteropObservable.ts", "node_modules/rxjs/src/internal/util/isAsyncIterable.ts", "node_modules/rxjs/src/internal/util/throwUnobservableError.ts", "node_modules/rxjs/src/internal/symbol/iterator.ts", "node_modules/rxjs/src/internal/util/isIterable.ts", "node_modules/rxjs/src/internal/util/isReadableStreamLike.ts", "node_modules/rxjs/src/internal/observable/innerFrom.ts", "node_modules/rxjs/src/internal/util/executeSchedule.ts", "node_modules/rxjs/src/internal/operators/observeOn.ts", "node_modules/rxjs/src/internal/operators/subscribeOn.ts", "node_modules/rxjs/src/internal/scheduled/scheduleObservable.ts", "node_modules/rxjs/src/internal/scheduled/schedulePromise.ts", "node_modules/rxjs/src/internal/scheduled/scheduleArray.ts", "node_modules/rxjs/src/internal/scheduled/scheduleIterable.ts", "node_modules/rxjs/src/internal/scheduled/scheduleAsyncIterable.ts", "node_modules/rxjs/src/internal/scheduled/scheduleReadableStreamLike.ts", "node_modules/rxjs/src/internal/scheduled/scheduled.ts", "node_modules/rxjs/src/internal/observable/from.ts", "node_modules/rxjs/src/internal/observable/of.ts", "node_modules/rxjs/src/internal/observable/throwError.ts", "node_modules/rxjs/src/internal/util/EmptyError.ts", "node_modules/rxjs/src/internal/util/isDate.ts", "node_modules/rxjs/src/internal/operators/map.ts", "node_modules/rxjs/src/internal/util/mapOneOrManyArgs.ts", "node_modules/rxjs/src/internal/util/argsArgArrayOrObject.ts", "node_modules/rxjs/src/internal/util/createObject.ts", "node_modules/rxjs/src/internal/observable/combineLatest.ts", "node_modules/rxjs/src/internal/operators/mergeInternals.ts", "node_modules/rxjs/src/internal/operators/mergeMap.ts", "node_modules/rxjs/src/internal/operators/mergeAll.ts", "node_modules/rxjs/src/internal/operators/concatAll.ts", "node_modules/rxjs/src/internal/observable/concat.ts", "node_modules/rxjs/src/internal/observable/defer.ts", "node_modules/rxjs/src/internal/observable/fromEvent.ts", "node_modules/rxjs/src/internal/observable/fromEventPattern.ts", "node_modules/rxjs/src/internal/observable/timer.ts", "node_modules/rxjs/src/internal/observable/merge.ts", "node_modules/rxjs/src/internal/observable/never.ts", "node_modules/rxjs/src/internal/util/argsOrArgArray.ts", "node_modules/rxjs/src/internal/operators/filter.ts", "node_modules/rxjs/src/internal/observable/zip.ts", "node_modules/rxjs/src/internal/operators/audit.ts", "node_modules/rxjs/src/internal/operators/auditTime.ts", "node_modules/rxjs/src/internal/operators/bufferCount.ts", "node_modules/rxjs/src/internal/operators/catchError.ts", "node_modules/rxjs/src/internal/operators/scanInternals.ts", "node_modules/rxjs/src/internal/operators/combineLatest.ts", "node_modules/rxjs/src/internal/operators/combineLatestWith.ts", "node_modules/rxjs/src/internal/operators/debounce.ts", "node_modules/rxjs/src/internal/operators/debounceTime.ts", "node_modules/rxjs/src/internal/operators/defaultIfEmpty.ts", "node_modules/rxjs/src/internal/operators/take.ts", "node_modules/rxjs/src/internal/operators/ignoreElements.ts", "node_modules/rxjs/src/internal/operators/mapTo.ts", "node_modules/rxjs/src/internal/operators/delayWhen.ts", "node_modules/rxjs/src/internal/operators/delay.ts", "node_modules/rxjs/src/internal/operators/distinct.ts", "node_modules/rxjs/src/internal/operators/distinctUntilChanged.ts", "node_modules/rxjs/src/internal/operators/distinctUntilKeyChanged.ts", "node_modules/rxjs/src/internal/operators/throwIfEmpty.ts", "node_modules/rxjs/src/internal/operators/endWith.ts", "node_modules/rxjs/src/internal/operators/exhaustMap.ts", "node_modules/rxjs/src/internal/operators/finalize.ts", "node_modules/rxjs/src/internal/operators/first.ts", "node_modules/rxjs/src/internal/operators/takeLast.ts", "node_modules/rxjs/src/internal/operators/merge.ts", "node_modules/rxjs/src/internal/operators/mergeWith.ts", "node_modules/rxjs/src/internal/operators/repeat.ts", "node_modules/rxjs/src/internal/operators/scan.ts", "node_modules/rxjs/src/internal/operators/share.ts", "node_modules/rxjs/src/internal/operators/shareReplay.ts", "node_modules/rxjs/src/internal/operators/skip.ts", "node_modules/rxjs/src/internal/operators/skipUntil.ts", "node_modules/rxjs/src/internal/operators/startWith.ts", "node_modules/rxjs/src/internal/operators/switchMap.ts", "node_modules/rxjs/src/internal/operators/takeUntil.ts", "node_modules/rxjs/src/internal/operators/takeWhile.ts", "node_modules/rxjs/src/internal/operators/tap.ts", "node_modules/rxjs/src/internal/operators/throttle.ts", "node_modules/rxjs/src/internal/operators/throttleTime.ts", "node_modules/rxjs/src/internal/operators/withLatestFrom.ts", "node_modules/rxjs/src/internal/operators/zip.ts", "node_modules/rxjs/src/internal/operators/zipWith.ts", "src/templates/assets/javascripts/browser/document/index.ts", "src/templates/assets/javascripts/browser/element/_/index.ts", "src/templates/assets/javascripts/browser/element/focus/index.ts", "src/templates/assets/javascripts/browser/element/hover/index.ts", "src/templates/assets/javascripts/utilities/h/index.ts", "src/templates/assets/javascripts/utilities/round/index.ts", "src/templates/assets/javascripts/browser/script/index.ts", "src/templates/assets/javascripts/browser/element/size/_/index.ts", "src/templates/assets/javascripts/browser/element/size/content/index.ts", "src/templates/assets/javascripts/browser/element/offset/_/index.ts", "src/templates/assets/javascripts/browser/element/offset/content/index.ts", "src/templates/assets/javascripts/browser/element/visibility/index.ts", "src/templates/assets/javascripts/browser/toggle/index.ts", "src/templates/assets/javascripts/browser/keyboard/index.ts", "src/templates/assets/javascripts/browser/location/_/index.ts", "src/templates/assets/javascripts/browser/location/hash/index.ts", "src/templates/assets/javascripts/browser/media/index.ts", "src/templates/assets/javascripts/browser/request/index.ts", "src/templates/assets/javascripts/browser/viewport/offset/index.ts", "src/templates/assets/javascripts/browser/viewport/size/index.ts", "src/templates/assets/javascripts/browser/viewport/_/index.ts", "src/templates/assets/javascripts/browser/viewport/at/index.ts", "src/templates/assets/javascripts/browser/worker/index.ts", "src/templates/assets/javascripts/_/index.ts", "src/templates/assets/javascripts/components/_/index.ts", "src/templates/assets/javascripts/components/announce/index.ts", "src/templates/assets/javascripts/components/consent/index.ts", "src/templates/assets/javascripts/templates/tooltip/index.tsx", "src/templates/assets/javascripts/templates/annotation/index.tsx", "src/templates/assets/javascripts/templates/clipboard/index.tsx", "src/templates/assets/javascripts/templates/search/index.tsx", "src/templates/assets/javascripts/templates/source/index.tsx", "src/templates/assets/javascripts/templates/tabbed/index.tsx", "src/templates/assets/javascripts/templates/table/index.tsx", "src/templates/assets/javascripts/templates/version/index.tsx", "src/templates/assets/javascripts/components/tooltip2/index.ts", "src/templates/assets/javascripts/components/content/annotation/_/index.ts", "src/templates/assets/javascripts/components/content/annotation/list/index.ts", "src/templates/assets/javascripts/components/content/annotation/block/index.ts", "src/templates/assets/javascripts/components/content/code/_/index.ts", "src/templates/assets/javascripts/components/content/details/index.ts", "src/templates/assets/javascripts/components/content/link/index.ts", "src/templates/assets/javascripts/components/content/mermaid/index.css", "src/templates/assets/javascripts/components/content/mermaid/index.ts", "src/templates/assets/javascripts/components/content/table/index.ts", "src/templates/assets/javascripts/components/content/tabs/index.ts", "src/templates/assets/javascripts/components/content/_/index.ts", "src/templates/assets/javascripts/components/dialog/index.ts", "src/templates/assets/javascripts/components/tooltip/index.ts", "src/templates/assets/javascripts/components/header/_/index.ts", "src/templates/assets/javascripts/components/header/title/index.ts", "src/templates/assets/javascripts/components/main/index.ts", "src/templates/assets/javascripts/components/palette/index.ts", "src/templates/assets/javascripts/components/progress/index.ts", "src/templates/assets/javascripts/integrations/sitemap/index.ts", "src/templates/assets/javascripts/integrations/alternate/index.ts", "src/templates/assets/javascripts/integrations/clipboard/index.ts", "src/templates/assets/javascripts/integrations/instant/index.ts", "src/templates/assets/javascripts/integrations/search/highlighter/index.ts", "src/templates/assets/javascripts/integrations/search/worker/message/index.ts", "src/templates/assets/javascripts/integrations/search/worker/_/index.ts", "src/templates/assets/javascripts/integrations/version/findurl/index.ts", "src/templates/assets/javascripts/integrations/version/index.ts", "src/templates/assets/javascripts/components/search/query/index.ts", "src/templates/assets/javascripts/components/search/result/index.ts", "src/templates/assets/javascripts/components/search/share/index.ts", "src/templates/assets/javascripts/components/search/suggest/index.ts", "src/templates/assets/javascripts/components/search/_/index.ts", "src/templates/assets/javascripts/components/search/highlight/index.ts", "src/templates/assets/javascripts/components/sidebar/index.ts", "src/templates/assets/javascripts/components/source/facts/github/index.ts", "src/templates/assets/javascripts/components/source/facts/gitlab/index.ts", "src/templates/assets/javascripts/components/source/facts/_/index.ts", "src/templates/assets/javascripts/components/source/_/index.ts", "src/templates/assets/javascripts/components/tabs/index.ts", "src/templates/assets/javascripts/components/toc/index.ts", "src/templates/assets/javascripts/components/top/index.ts", "src/templates/assets/javascripts/patches/ellipsis/index.ts", "src/templates/assets/javascripts/patches/indeterminate/index.ts", "src/templates/assets/javascripts/patches/scrollfix/index.ts", "src/templates/assets/javascripts/patches/scrolllock/index.ts", "src/templates/assets/javascripts/polyfills/index.ts"], + "sourcesContent": ["(function (global, factory) {\n typeof exports === 'object' && typeof module !== 'undefined' ? factory() :\n typeof define === 'function' && define.amd ? define(factory) :\n (factory());\n}(this, (function () { 'use strict';\n\n /**\n * Applies the :focus-visible polyfill at the given scope.\n * A scope in this case is either the top-level Document or a Shadow Root.\n *\n * @param {(Document|ShadowRoot)} scope\n * @see https://github.com/WICG/focus-visible\n */\n function applyFocusVisiblePolyfill(scope) {\n var hadKeyboardEvent = true;\n var hadFocusVisibleRecently = false;\n var hadFocusVisibleRecentlyTimeout = null;\n\n var inputTypesAllowlist = {\n text: true,\n search: true,\n url: true,\n tel: true,\n email: true,\n password: true,\n number: true,\n date: true,\n month: true,\n week: true,\n time: true,\n datetime: true,\n 'datetime-local': true\n };\n\n /**\n * Helper function for legacy browsers and iframes which sometimes focus\n * elements like document, body, and non-interactive SVG.\n * @param {Element} el\n */\n function isValidFocusTarget(el) {\n if (\n el &&\n el !== document &&\n el.nodeName !== 'HTML' &&\n el.nodeName !== 'BODY' &&\n 'classList' in el &&\n 'contains' in el.classList\n ) {\n return true;\n }\n return false;\n }\n\n /**\n * Computes whether the given element should automatically trigger the\n * `focus-visible` class being added, i.e. whether it should always match\n * `:focus-visible` when focused.\n * @param {Element} el\n * @return {boolean}\n */\n function focusTriggersKeyboardModality(el) {\n var type = el.type;\n var tagName = el.tagName;\n\n if (tagName === 'INPUT' && inputTypesAllowlist[type] && !el.readOnly) {\n return true;\n }\n\n if (tagName === 'TEXTAREA' && !el.readOnly) {\n return true;\n }\n\n if (el.isContentEditable) {\n return true;\n }\n\n return false;\n }\n\n /**\n * Add the `focus-visible` class to the given element if it was not added by\n * the author.\n * @param {Element} el\n */\n function addFocusVisibleClass(el) {\n if (el.classList.contains('focus-visible')) {\n return;\n }\n el.classList.add('focus-visible');\n el.setAttribute('data-focus-visible-added', '');\n }\n\n /**\n * Remove the `focus-visible` class from the given element if it was not\n * originally added by the author.\n * @param {Element} el\n */\n function removeFocusVisibleClass(el) {\n if (!el.hasAttribute('data-focus-visible-added')) {\n return;\n }\n el.classList.remove('focus-visible');\n el.removeAttribute('data-focus-visible-added');\n }\n\n /**\n * If the most recent user interaction was via the keyboard;\n * and the key press did not include a meta, alt/option, or control key;\n * then the modality is keyboard. Otherwise, the modality is not keyboard.\n * Apply `focus-visible` to any current active element and keep track\n * of our keyboard modality state with `hadKeyboardEvent`.\n * @param {KeyboardEvent} e\n */\n function onKeyDown(e) {\n if (e.metaKey || e.altKey || e.ctrlKey) {\n return;\n }\n\n if (isValidFocusTarget(scope.activeElement)) {\n addFocusVisibleClass(scope.activeElement);\n }\n\n hadKeyboardEvent = true;\n }\n\n /**\n * If at any point a user clicks with a pointing device, ensure that we change\n * the modality away from keyboard.\n * This avoids the situation where a user presses a key on an already focused\n * element, and then clicks on a different element, focusing it with a\n * pointing device, while we still think we're in keyboard modality.\n * @param {Event} e\n */\n function onPointerDown(e) {\n hadKeyboardEvent = false;\n }\n\n /**\n * On `focus`, add the `focus-visible` class to the target if:\n * - the target received focus as a result of keyboard navigation, or\n * - the event target is an element that will likely require interaction\n * via the keyboard (e.g. a text box)\n * @param {Event} e\n */\n function onFocus(e) {\n // Prevent IE from focusing the document or HTML element.\n if (!isValidFocusTarget(e.target)) {\n return;\n }\n\n if (hadKeyboardEvent || focusTriggersKeyboardModality(e.target)) {\n addFocusVisibleClass(e.target);\n }\n }\n\n /**\n * On `blur`, remove the `focus-visible` class from the target.\n * @param {Event} e\n */\n function onBlur(e) {\n if (!isValidFocusTarget(e.target)) {\n return;\n }\n\n if (\n e.target.classList.contains('focus-visible') ||\n e.target.hasAttribute('data-focus-visible-added')\n ) {\n // To detect a tab/window switch, we look for a blur event followed\n // rapidly by a visibility change.\n // If we don't see a visibility change within 100ms, it's probably a\n // regular focus change.\n hadFocusVisibleRecently = true;\n window.clearTimeout(hadFocusVisibleRecentlyTimeout);\n hadFocusVisibleRecentlyTimeout = window.setTimeout(function() {\n hadFocusVisibleRecently = false;\n }, 100);\n removeFocusVisibleClass(e.target);\n }\n }\n\n /**\n * If the user changes tabs, keep track of whether or not the previously\n * focused element had .focus-visible.\n * @param {Event} e\n */\n function onVisibilityChange(e) {\n if (document.visibilityState === 'hidden') {\n // If the tab becomes active again, the browser will handle calling focus\n // on the element (Safari actually calls it twice).\n // If this tab change caused a blur on an element with focus-visible,\n // re-apply the class when the user switches back to the tab.\n if (hadFocusVisibleRecently) {\n hadKeyboardEvent = true;\n }\n addInitialPointerMoveListeners();\n }\n }\n\n /**\n * Add a group of listeners to detect usage of any pointing devices.\n * These listeners will be added when the polyfill first loads, and anytime\n * the window is blurred, so that they are active when the window regains\n * focus.\n */\n function addInitialPointerMoveListeners() {\n document.addEventListener('mousemove', onInitialPointerMove);\n document.addEventListener('mousedown', onInitialPointerMove);\n document.addEventListener('mouseup', onInitialPointerMove);\n document.addEventListener('pointermove', onInitialPointerMove);\n document.addEventListener('pointerdown', onInitialPointerMove);\n document.addEventListener('pointerup', onInitialPointerMove);\n document.addEventListener('touchmove', onInitialPointerMove);\n document.addEventListener('touchstart', onInitialPointerMove);\n document.addEventListener('touchend', onInitialPointerMove);\n }\n\n function removeInitialPointerMoveListeners() {\n document.removeEventListener('mousemove', onInitialPointerMove);\n document.removeEventListener('mousedown', onInitialPointerMove);\n document.removeEventListener('mouseup', onInitialPointerMove);\n document.removeEventListener('pointermove', onInitialPointerMove);\n document.removeEventListener('pointerdown', onInitialPointerMove);\n document.removeEventListener('pointerup', onInitialPointerMove);\n document.removeEventListener('touchmove', onInitialPointerMove);\n document.removeEventListener('touchstart', onInitialPointerMove);\n document.removeEventListener('touchend', onInitialPointerMove);\n }\n\n /**\n * When the polfyill first loads, assume the user is in keyboard modality.\n * If any event is received from a pointing device (e.g. mouse, pointer,\n * touch), turn off keyboard modality.\n * This accounts for situations where focus enters the page from the URL bar.\n * @param {Event} e\n */\n function onInitialPointerMove(e) {\n // Work around a Safari quirk that fires a mousemove on whenever the\n // window blurs, even if you're tabbing out of the page. \u00AF\\_(\u30C4)_/\u00AF\n if (e.target.nodeName && e.target.nodeName.toLowerCase() === 'html') {\n return;\n }\n\n hadKeyboardEvent = false;\n removeInitialPointerMoveListeners();\n }\n\n // For some kinds of state, we are interested in changes at the global scope\n // only. For example, global pointer input, global key presses and global\n // visibility change should affect the state at every scope:\n document.addEventListener('keydown', onKeyDown, true);\n document.addEventListener('mousedown', onPointerDown, true);\n document.addEventListener('pointerdown', onPointerDown, true);\n document.addEventListener('touchstart', onPointerDown, true);\n document.addEventListener('visibilitychange', onVisibilityChange, true);\n\n addInitialPointerMoveListeners();\n\n // For focus and blur, we specifically care about state changes in the local\n // scope. This is because focus / blur events that originate from within a\n // shadow root are not re-dispatched from the host element if it was already\n // the active element in its own scope:\n scope.addEventListener('focus', onFocus, true);\n scope.addEventListener('blur', onBlur, true);\n\n // We detect that a node is a ShadowRoot by ensuring that it is a\n // DocumentFragment and also has a host property. This check covers native\n // implementation and polyfill implementation transparently. If we only cared\n // about the native implementation, we could just check if the scope was\n // an instance of a ShadowRoot.\n if (scope.nodeType === Node.DOCUMENT_FRAGMENT_NODE && scope.host) {\n // Since a ShadowRoot is a special kind of DocumentFragment, it does not\n // have a root element to add a class to. So, we add this attribute to the\n // host element instead:\n scope.host.setAttribute('data-js-focus-visible', '');\n } else if (scope.nodeType === Node.DOCUMENT_NODE) {\n document.documentElement.classList.add('js-focus-visible');\n document.documentElement.setAttribute('data-js-focus-visible', '');\n }\n }\n\n // It is important to wrap all references to global window and document in\n // these checks to support server-side rendering use cases\n // @see https://github.com/WICG/focus-visible/issues/199\n if (typeof window !== 'undefined' && typeof document !== 'undefined') {\n // Make the polyfill helper globally available. This can be used as a signal\n // to interested libraries that wish to coordinate with the polyfill for e.g.,\n // applying the polyfill to a shadow root:\n window.applyFocusVisiblePolyfill = applyFocusVisiblePolyfill;\n\n // Notify interested libraries of the polyfill's presence, in case the\n // polyfill was loaded lazily:\n var event;\n\n try {\n event = new CustomEvent('focus-visible-polyfill-ready');\n } catch (error) {\n // IE11 does not support using CustomEvent as a constructor directly:\n event = document.createEvent('CustomEvent');\n event.initCustomEvent('focus-visible-polyfill-ready', false, false, {});\n }\n\n window.dispatchEvent(event);\n }\n\n if (typeof document !== 'undefined') {\n // Apply the polyfill to the global document, so that no JavaScript\n // coordination is required to use the polyfill in the top-level document:\n applyFocusVisiblePolyfill(document);\n }\n\n})));\n", "/*!\n * escape-html\n * Copyright(c) 2012-2013 TJ Holowaychuk\n * Copyright(c) 2015 Andreas Lubbe\n * Copyright(c) 2015 Tiancheng \"Timothy\" Gu\n * MIT Licensed\n */\n\n'use strict';\n\n/**\n * Module variables.\n * @private\n */\n\nvar matchHtmlRegExp = /[\"'&<>]/;\n\n/**\n * Module exports.\n * @public\n */\n\nmodule.exports = escapeHtml;\n\n/**\n * Escape special characters in the given string of html.\n *\n * @param {string} string The string to escape for inserting into HTML\n * @return {string}\n * @public\n */\n\nfunction escapeHtml(string) {\n var str = '' + string;\n var match = matchHtmlRegExp.exec(str);\n\n if (!match) {\n return str;\n }\n\n var escape;\n var html = '';\n var index = 0;\n var lastIndex = 0;\n\n for (index = match.index; index < str.length; index++) {\n switch (str.charCodeAt(index)) {\n case 34: // \"\n escape = '"';\n break;\n case 38: // &\n escape = '&';\n break;\n case 39: // '\n escape = ''';\n break;\n case 60: // <\n escape = '<';\n break;\n case 62: // >\n escape = '>';\n break;\n default:\n continue;\n }\n\n if (lastIndex !== index) {\n html += str.substring(lastIndex, index);\n }\n\n lastIndex = index + 1;\n html += escape;\n }\n\n return lastIndex !== index\n ? html + str.substring(lastIndex, index)\n : html;\n}\n", "/*!\n * clipboard.js v2.0.11\n * https://clipboardjs.com/\n *\n * Licensed MIT \u00A9 Zeno Rocha\n */\n(function webpackUniversalModuleDefinition(root, factory) {\n\tif(typeof exports === 'object' && typeof module === 'object')\n\t\tmodule.exports = factory();\n\telse if(typeof define === 'function' && define.amd)\n\t\tdefine([], factory);\n\telse if(typeof exports === 'object')\n\t\texports[\"ClipboardJS\"] = factory();\n\telse\n\t\troot[\"ClipboardJS\"] = factory();\n})(this, function() {\nreturn /******/ (function() { // webpackBootstrap\n/******/ \tvar __webpack_modules__ = ({\n\n/***/ 686:\n/***/ (function(__unused_webpack_module, __webpack_exports__, __webpack_require__) {\n\n\"use strict\";\n\n// EXPORTS\n__webpack_require__.d(__webpack_exports__, {\n \"default\": function() { return /* binding */ clipboard; }\n});\n\n// EXTERNAL MODULE: ./node_modules/tiny-emitter/index.js\nvar tiny_emitter = __webpack_require__(279);\nvar tiny_emitter_default = /*#__PURE__*/__webpack_require__.n(tiny_emitter);\n// EXTERNAL MODULE: ./node_modules/good-listener/src/listen.js\nvar listen = __webpack_require__(370);\nvar listen_default = /*#__PURE__*/__webpack_require__.n(listen);\n// EXTERNAL MODULE: ./node_modules/select/src/select.js\nvar src_select = __webpack_require__(817);\nvar select_default = /*#__PURE__*/__webpack_require__.n(src_select);\n;// CONCATENATED MODULE: ./src/common/command.js\n/**\n * Executes a given operation type.\n * @param {String} type\n * @return {Boolean}\n */\nfunction command(type) {\n try {\n return document.execCommand(type);\n } catch (err) {\n return false;\n }\n}\n;// CONCATENATED MODULE: ./src/actions/cut.js\n\n\n/**\n * Cut action wrapper.\n * @param {String|HTMLElement} target\n * @return {String}\n */\n\nvar ClipboardActionCut = function ClipboardActionCut(target) {\n var selectedText = select_default()(target);\n command('cut');\n return selectedText;\n};\n\n/* harmony default export */ var actions_cut = (ClipboardActionCut);\n;// CONCATENATED MODULE: ./src/common/create-fake-element.js\n/**\n * Creates a fake textarea element with a value.\n * @param {String} value\n * @return {HTMLElement}\n */\nfunction createFakeElement(value) {\n var isRTL = document.documentElement.getAttribute('dir') === 'rtl';\n var fakeElement = document.createElement('textarea'); // Prevent zooming on iOS\n\n fakeElement.style.fontSize = '12pt'; // Reset box model\n\n fakeElement.style.border = '0';\n fakeElement.style.padding = '0';\n fakeElement.style.margin = '0'; // Move element out of screen horizontally\n\n fakeElement.style.position = 'absolute';\n fakeElement.style[isRTL ? 'right' : 'left'] = '-9999px'; // Move element to the same position vertically\n\n var yPosition = window.pageYOffset || document.documentElement.scrollTop;\n fakeElement.style.top = \"\".concat(yPosition, \"px\");\n fakeElement.setAttribute('readonly', '');\n fakeElement.value = value;\n return fakeElement;\n}\n;// CONCATENATED MODULE: ./src/actions/copy.js\n\n\n\n/**\n * Create fake copy action wrapper using a fake element.\n * @param {String} target\n * @param {Object} options\n * @return {String}\n */\n\nvar fakeCopyAction = function fakeCopyAction(value, options) {\n var fakeElement = createFakeElement(value);\n options.container.appendChild(fakeElement);\n var selectedText = select_default()(fakeElement);\n command('copy');\n fakeElement.remove();\n return selectedText;\n};\n/**\n * Copy action wrapper.\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @return {String}\n */\n\n\nvar ClipboardActionCopy = function ClipboardActionCopy(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n container: document.body\n };\n var selectedText = '';\n\n if (typeof target === 'string') {\n selectedText = fakeCopyAction(target, options);\n } else if (target instanceof HTMLInputElement && !['text', 'search', 'url', 'tel', 'password'].includes(target === null || target === void 0 ? void 0 : target.type)) {\n // If input type doesn't support `setSelectionRange`. Simulate it. https://developer.mozilla.org/en-US/docs/Web/API/HTMLInputElement/setSelectionRange\n selectedText = fakeCopyAction(target.value, options);\n } else {\n selectedText = select_default()(target);\n command('copy');\n }\n\n return selectedText;\n};\n\n/* harmony default export */ var actions_copy = (ClipboardActionCopy);\n;// CONCATENATED MODULE: ./src/actions/default.js\nfunction _typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { _typeof = function _typeof(obj) { return typeof obj; }; } else { _typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return _typeof(obj); }\n\n\n\n/**\n * Inner function which performs selection from either `text` or `target`\n * properties and then executes copy or cut operations.\n * @param {Object} options\n */\n\nvar ClipboardActionDefault = function ClipboardActionDefault() {\n var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n // Defines base properties passed from constructor.\n var _options$action = options.action,\n action = _options$action === void 0 ? 'copy' : _options$action,\n container = options.container,\n target = options.target,\n text = options.text; // Sets the `action` to be performed which can be either 'copy' or 'cut'.\n\n if (action !== 'copy' && action !== 'cut') {\n throw new Error('Invalid \"action\" value, use either \"copy\" or \"cut\"');\n } // Sets the `target` property using an element that will be have its content copied.\n\n\n if (target !== undefined) {\n if (target && _typeof(target) === 'object' && target.nodeType === 1) {\n if (action === 'copy' && target.hasAttribute('disabled')) {\n throw new Error('Invalid \"target\" attribute. Please use \"readonly\" instead of \"disabled\" attribute');\n }\n\n if (action === 'cut' && (target.hasAttribute('readonly') || target.hasAttribute('disabled'))) {\n throw new Error('Invalid \"target\" attribute. You can\\'t cut text from elements with \"readonly\" or \"disabled\" attributes');\n }\n } else {\n throw new Error('Invalid \"target\" value, use a valid Element');\n }\n } // Define selection strategy based on `text` property.\n\n\n if (text) {\n return actions_copy(text, {\n container: container\n });\n } // Defines which selection strategy based on `target` property.\n\n\n if (target) {\n return action === 'cut' ? actions_cut(target) : actions_copy(target, {\n container: container\n });\n }\n};\n\n/* harmony default export */ var actions_default = (ClipboardActionDefault);\n;// CONCATENATED MODULE: ./src/clipboard.js\nfunction clipboard_typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { clipboard_typeof = function _typeof(obj) { return typeof obj; }; } else { clipboard_typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return clipboard_typeof(obj); }\n\nfunction _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError(\"Cannot call a class as a function\"); } }\n\nfunction _defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if (\"value\" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } }\n\nfunction _createClass(Constructor, protoProps, staticProps) { if (protoProps) _defineProperties(Constructor.prototype, protoProps); if (staticProps) _defineProperties(Constructor, staticProps); return Constructor; }\n\nfunction _inherits(subClass, superClass) { if (typeof superClass !== \"function\" && superClass !== null) { throw new TypeError(\"Super expression must either be null or a function\"); } subClass.prototype = Object.create(superClass && superClass.prototype, { constructor: { value: subClass, writable: true, configurable: true } }); if (superClass) _setPrototypeOf(subClass, superClass); }\n\nfunction _setPrototypeOf(o, p) { _setPrototypeOf = Object.setPrototypeOf || function _setPrototypeOf(o, p) { o.__proto__ = p; return o; }; return _setPrototypeOf(o, p); }\n\nfunction _createSuper(Derived) { var hasNativeReflectConstruct = _isNativeReflectConstruct(); return function _createSuperInternal() { var Super = _getPrototypeOf(Derived), result; if (hasNativeReflectConstruct) { var NewTarget = _getPrototypeOf(this).constructor; result = Reflect.construct(Super, arguments, NewTarget); } else { result = Super.apply(this, arguments); } return _possibleConstructorReturn(this, result); }; }\n\nfunction _possibleConstructorReturn(self, call) { if (call && (clipboard_typeof(call) === \"object\" || typeof call === \"function\")) { return call; } return _assertThisInitialized(self); }\n\nfunction _assertThisInitialized(self) { if (self === void 0) { throw new ReferenceError(\"this hasn't been initialised - super() hasn't been called\"); } return self; }\n\nfunction _isNativeReflectConstruct() { if (typeof Reflect === \"undefined\" || !Reflect.construct) return false; if (Reflect.construct.sham) return false; if (typeof Proxy === \"function\") return true; try { Date.prototype.toString.call(Reflect.construct(Date, [], function () {})); return true; } catch (e) { return false; } }\n\nfunction _getPrototypeOf(o) { _getPrototypeOf = Object.setPrototypeOf ? Object.getPrototypeOf : function _getPrototypeOf(o) { return o.__proto__ || Object.getPrototypeOf(o); }; return _getPrototypeOf(o); }\n\n\n\n\n\n\n/**\n * Helper function to retrieve attribute value.\n * @param {String} suffix\n * @param {Element} element\n */\n\nfunction getAttributeValue(suffix, element) {\n var attribute = \"data-clipboard-\".concat(suffix);\n\n if (!element.hasAttribute(attribute)) {\n return;\n }\n\n return element.getAttribute(attribute);\n}\n/**\n * Base class which takes one or more elements, adds event listeners to them,\n * and instantiates a new `ClipboardAction` on each click.\n */\n\n\nvar Clipboard = /*#__PURE__*/function (_Emitter) {\n _inherits(Clipboard, _Emitter);\n\n var _super = _createSuper(Clipboard);\n\n /**\n * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n * @param {Object} options\n */\n function Clipboard(trigger, options) {\n var _this;\n\n _classCallCheck(this, Clipboard);\n\n _this = _super.call(this);\n\n _this.resolveOptions(options);\n\n _this.listenClick(trigger);\n\n return _this;\n }\n /**\n * Defines if attributes would be resolved using internal setter functions\n * or custom functions that were passed in the constructor.\n * @param {Object} options\n */\n\n\n _createClass(Clipboard, [{\n key: \"resolveOptions\",\n value: function resolveOptions() {\n var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n this.action = typeof options.action === 'function' ? options.action : this.defaultAction;\n this.target = typeof options.target === 'function' ? options.target : this.defaultTarget;\n this.text = typeof options.text === 'function' ? options.text : this.defaultText;\n this.container = clipboard_typeof(options.container) === 'object' ? options.container : document.body;\n }\n /**\n * Adds a click event listener to the passed trigger.\n * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n */\n\n }, {\n key: \"listenClick\",\n value: function listenClick(trigger) {\n var _this2 = this;\n\n this.listener = listen_default()(trigger, 'click', function (e) {\n return _this2.onClick(e);\n });\n }\n /**\n * Defines a new `ClipboardAction` on each click event.\n * @param {Event} e\n */\n\n }, {\n key: \"onClick\",\n value: function onClick(e) {\n var trigger = e.delegateTarget || e.currentTarget;\n var action = this.action(trigger) || 'copy';\n var text = actions_default({\n action: action,\n container: this.container,\n target: this.target(trigger),\n text: this.text(trigger)\n }); // Fires an event based on the copy operation result.\n\n this.emit(text ? 'success' : 'error', {\n action: action,\n text: text,\n trigger: trigger,\n clearSelection: function clearSelection() {\n if (trigger) {\n trigger.focus();\n }\n\n window.getSelection().removeAllRanges();\n }\n });\n }\n /**\n * Default `action` lookup function.\n * @param {Element} trigger\n */\n\n }, {\n key: \"defaultAction\",\n value: function defaultAction(trigger) {\n return getAttributeValue('action', trigger);\n }\n /**\n * Default `target` lookup function.\n * @param {Element} trigger\n */\n\n }, {\n key: \"defaultTarget\",\n value: function defaultTarget(trigger) {\n var selector = getAttributeValue('target', trigger);\n\n if (selector) {\n return document.querySelector(selector);\n }\n }\n /**\n * Allow fire programmatically a copy action\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @returns Text copied.\n */\n\n }, {\n key: \"defaultText\",\n\n /**\n * Default `text` lookup function.\n * @param {Element} trigger\n */\n value: function defaultText(trigger) {\n return getAttributeValue('text', trigger);\n }\n /**\n * Destroy lifecycle.\n */\n\n }, {\n key: \"destroy\",\n value: function destroy() {\n this.listener.destroy();\n }\n }], [{\n key: \"copy\",\n value: function copy(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n container: document.body\n };\n return actions_copy(target, options);\n }\n /**\n * Allow fire programmatically a cut action\n * @param {String|HTMLElement} target\n * @returns Text cutted.\n */\n\n }, {\n key: \"cut\",\n value: function cut(target) {\n return actions_cut(target);\n }\n /**\n * Returns the support of the given action, or all actions if no action is\n * given.\n * @param {String} [action]\n */\n\n }, {\n key: \"isSupported\",\n value: function isSupported() {\n var action = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : ['copy', 'cut'];\n var actions = typeof action === 'string' ? [action] : action;\n var support = !!document.queryCommandSupported;\n actions.forEach(function (action) {\n support = support && !!document.queryCommandSupported(action);\n });\n return support;\n }\n }]);\n\n return Clipboard;\n}((tiny_emitter_default()));\n\n/* harmony default export */ var clipboard = (Clipboard);\n\n/***/ }),\n\n/***/ 828:\n/***/ (function(module) {\n\nvar DOCUMENT_NODE_TYPE = 9;\n\n/**\n * A polyfill for Element.matches()\n */\nif (typeof Element !== 'undefined' && !Element.prototype.matches) {\n var proto = Element.prototype;\n\n proto.matches = proto.matchesSelector ||\n proto.mozMatchesSelector ||\n proto.msMatchesSelector ||\n proto.oMatchesSelector ||\n proto.webkitMatchesSelector;\n}\n\n/**\n * Finds the closest parent that matches a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @return {Function}\n */\nfunction closest (element, selector) {\n while (element && element.nodeType !== DOCUMENT_NODE_TYPE) {\n if (typeof element.matches === 'function' &&\n element.matches(selector)) {\n return element;\n }\n element = element.parentNode;\n }\n}\n\nmodule.exports = closest;\n\n\n/***/ }),\n\n/***/ 438:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar closest = __webpack_require__(828);\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction _delegate(element, selector, type, callback, useCapture) {\n var listenerFn = listener.apply(this, arguments);\n\n element.addEventListener(type, listenerFn, useCapture);\n\n return {\n destroy: function() {\n element.removeEventListener(type, listenerFn, useCapture);\n }\n }\n}\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element|String|Array} [elements]\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction delegate(elements, selector, type, callback, useCapture) {\n // Handle the regular Element usage\n if (typeof elements.addEventListener === 'function') {\n return _delegate.apply(null, arguments);\n }\n\n // Handle Element-less usage, it defaults to global delegation\n if (typeof type === 'function') {\n // Use `document` as the first parameter, then apply arguments\n // This is a short way to .unshift `arguments` without running into deoptimizations\n return _delegate.bind(null, document).apply(null, arguments);\n }\n\n // Handle Selector-based usage\n if (typeof elements === 'string') {\n elements = document.querySelectorAll(elements);\n }\n\n // Handle Array-like based usage\n return Array.prototype.map.call(elements, function (element) {\n return _delegate(element, selector, type, callback, useCapture);\n });\n}\n\n/**\n * Finds closest match and invokes callback.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Function}\n */\nfunction listener(element, selector, type, callback) {\n return function(e) {\n e.delegateTarget = closest(e.target, selector);\n\n if (e.delegateTarget) {\n callback.call(element, e);\n }\n }\n}\n\nmodule.exports = delegate;\n\n\n/***/ }),\n\n/***/ 879:\n/***/ (function(__unused_webpack_module, exports) {\n\n/**\n * Check if argument is a HTML element.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.node = function(value) {\n return value !== undefined\n && value instanceof HTMLElement\n && value.nodeType === 1;\n};\n\n/**\n * Check if argument is a list of HTML elements.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.nodeList = function(value) {\n var type = Object.prototype.toString.call(value);\n\n return value !== undefined\n && (type === '[object NodeList]' || type === '[object HTMLCollection]')\n && ('length' in value)\n && (value.length === 0 || exports.node(value[0]));\n};\n\n/**\n * Check if argument is a string.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.string = function(value) {\n return typeof value === 'string'\n || value instanceof String;\n};\n\n/**\n * Check if argument is a function.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.fn = function(value) {\n var type = Object.prototype.toString.call(value);\n\n return type === '[object Function]';\n};\n\n\n/***/ }),\n\n/***/ 370:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar is = __webpack_require__(879);\nvar delegate = __webpack_require__(438);\n\n/**\n * Validates all params and calls the right\n * listener function based on its target type.\n *\n * @param {String|HTMLElement|HTMLCollection|NodeList} target\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listen(target, type, callback) {\n if (!target && !type && !callback) {\n throw new Error('Missing required arguments');\n }\n\n if (!is.string(type)) {\n throw new TypeError('Second argument must be a String');\n }\n\n if (!is.fn(callback)) {\n throw new TypeError('Third argument must be a Function');\n }\n\n if (is.node(target)) {\n return listenNode(target, type, callback);\n }\n else if (is.nodeList(target)) {\n return listenNodeList(target, type, callback);\n }\n else if (is.string(target)) {\n return listenSelector(target, type, callback);\n }\n else {\n throw new TypeError('First argument must be a String, HTMLElement, HTMLCollection, or NodeList');\n }\n}\n\n/**\n * Adds an event listener to a HTML element\n * and returns a remove listener function.\n *\n * @param {HTMLElement} node\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNode(node, type, callback) {\n node.addEventListener(type, callback);\n\n return {\n destroy: function() {\n node.removeEventListener(type, callback);\n }\n }\n}\n\n/**\n * Add an event listener to a list of HTML elements\n * and returns a remove listener function.\n *\n * @param {NodeList|HTMLCollection} nodeList\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNodeList(nodeList, type, callback) {\n Array.prototype.forEach.call(nodeList, function(node) {\n node.addEventListener(type, callback);\n });\n\n return {\n destroy: function() {\n Array.prototype.forEach.call(nodeList, function(node) {\n node.removeEventListener(type, callback);\n });\n }\n }\n}\n\n/**\n * Add an event listener to a selector\n * and returns a remove listener function.\n *\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenSelector(selector, type, callback) {\n return delegate(document.body, selector, type, callback);\n}\n\nmodule.exports = listen;\n\n\n/***/ }),\n\n/***/ 817:\n/***/ (function(module) {\n\nfunction select(element) {\n var selectedText;\n\n if (element.nodeName === 'SELECT') {\n element.focus();\n\n selectedText = element.value;\n }\n else if (element.nodeName === 'INPUT' || element.nodeName === 'TEXTAREA') {\n var isReadOnly = element.hasAttribute('readonly');\n\n if (!isReadOnly) {\n element.setAttribute('readonly', '');\n }\n\n element.select();\n element.setSelectionRange(0, element.value.length);\n\n if (!isReadOnly) {\n element.removeAttribute('readonly');\n }\n\n selectedText = element.value;\n }\n else {\n if (element.hasAttribute('contenteditable')) {\n element.focus();\n }\n\n var selection = window.getSelection();\n var range = document.createRange();\n\n range.selectNodeContents(element);\n selection.removeAllRanges();\n selection.addRange(range);\n\n selectedText = selection.toString();\n }\n\n return selectedText;\n}\n\nmodule.exports = select;\n\n\n/***/ }),\n\n/***/ 279:\n/***/ (function(module) {\n\nfunction E () {\n // Keep this empty so it's easier to inherit from\n // (via https://github.com/lipsmack from https://github.com/scottcorgan/tiny-emitter/issues/3)\n}\n\nE.prototype = {\n on: function (name, callback, ctx) {\n var e = this.e || (this.e = {});\n\n (e[name] || (e[name] = [])).push({\n fn: callback,\n ctx: ctx\n });\n\n return this;\n },\n\n once: function (name, callback, ctx) {\n var self = this;\n function listener () {\n self.off(name, listener);\n callback.apply(ctx, arguments);\n };\n\n listener._ = callback\n return this.on(name, listener, ctx);\n },\n\n emit: function (name) {\n var data = [].slice.call(arguments, 1);\n var evtArr = ((this.e || (this.e = {}))[name] || []).slice();\n var i = 0;\n var len = evtArr.length;\n\n for (i; i < len; i++) {\n evtArr[i].fn.apply(evtArr[i].ctx, data);\n }\n\n return this;\n },\n\n off: function (name, callback) {\n var e = this.e || (this.e = {});\n var evts = e[name];\n var liveEvents = [];\n\n if (evts && callback) {\n for (var i = 0, len = evts.length; i < len; i++) {\n if (evts[i].fn !== callback && evts[i].fn._ !== callback)\n liveEvents.push(evts[i]);\n }\n }\n\n // Remove event from queue to prevent memory leak\n // Suggested by https://github.com/lazd\n // Ref: https://github.com/scottcorgan/tiny-emitter/commit/c6ebfaa9bc973b33d110a84a307742b7cf94c953#commitcomment-5024910\n\n (liveEvents.length)\n ? e[name] = liveEvents\n : delete e[name];\n\n return this;\n }\n};\n\nmodule.exports = E;\nmodule.exports.TinyEmitter = E;\n\n\n/***/ })\n\n/******/ \t});\n/************************************************************************/\n/******/ \t// The module cache\n/******/ \tvar __webpack_module_cache__ = {};\n/******/ \t\n/******/ \t// The require function\n/******/ \tfunction __webpack_require__(moduleId) {\n/******/ \t\t// Check if module is in cache\n/******/ \t\tif(__webpack_module_cache__[moduleId]) {\n/******/ \t\t\treturn __webpack_module_cache__[moduleId].exports;\n/******/ \t\t}\n/******/ \t\t// Create a new module (and put it into the cache)\n/******/ \t\tvar module = __webpack_module_cache__[moduleId] = {\n/******/ \t\t\t// no module.id needed\n/******/ \t\t\t// no module.loaded needed\n/******/ \t\t\texports: {}\n/******/ \t\t};\n/******/ \t\n/******/ \t\t// Execute the module function\n/******/ \t\t__webpack_modules__[moduleId](module, module.exports, __webpack_require__);\n/******/ \t\n/******/ \t\t// Return the exports of the module\n/******/ \t\treturn module.exports;\n/******/ \t}\n/******/ \t\n/************************************************************************/\n/******/ \t/* webpack/runtime/compat get default export */\n/******/ \t!function() {\n/******/ \t\t// getDefaultExport function for compatibility with non-harmony modules\n/******/ \t\t__webpack_require__.n = function(module) {\n/******/ \t\t\tvar getter = module && module.__esModule ?\n/******/ \t\t\t\tfunction() { return module['default']; } :\n/******/ \t\t\t\tfunction() { return module; };\n/******/ \t\t\t__webpack_require__.d(getter, { a: getter });\n/******/ \t\t\treturn getter;\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/define property getters */\n/******/ \t!function() {\n/******/ \t\t// define getter functions for harmony exports\n/******/ \t\t__webpack_require__.d = function(exports, definition) {\n/******/ \t\t\tfor(var key in definition) {\n/******/ \t\t\t\tif(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {\n/******/ \t\t\t\t\tObject.defineProperty(exports, key, { enumerable: true, get: definition[key] });\n/******/ \t\t\t\t}\n/******/ \t\t\t}\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/hasOwnProperty shorthand */\n/******/ \t!function() {\n/******/ \t\t__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }\n/******/ \t}();\n/******/ \t\n/************************************************************************/\n/******/ \t// module exports must be returned from runtime so entry inlining is disabled\n/******/ \t// startup\n/******/ \t// Load entry module and return exports\n/******/ \treturn __webpack_require__(686);\n/******/ })()\n.default;\n});", "/*\n * Copyright (c) 2016-2025 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport \"focus-visible\"\n\nimport {\n EMPTY,\n NEVER,\n Observable,\n Subject,\n defer,\n delay,\n filter,\n map,\n merge,\n mergeWith,\n shareReplay,\n switchMap\n} from \"rxjs\"\n\nimport { configuration, feature } from \"./_\"\nimport {\n at,\n getActiveElement,\n getOptionalElement,\n requestJSON,\n setLocation,\n setToggle,\n watchDocument,\n watchKeyboard,\n watchLocation,\n watchLocationTarget,\n watchMedia,\n watchPrint,\n watchScript,\n watchViewport\n} from \"./browser\"\nimport {\n getComponentElement,\n getComponentElements,\n mountAnnounce,\n mountBackToTop,\n mountConsent,\n mountContent,\n mountDialog,\n mountHeader,\n mountHeaderTitle,\n mountPalette,\n mountProgress,\n mountSearch,\n mountSearchHiglight,\n mountSidebar,\n mountSource,\n mountTableOfContents,\n mountTabs,\n watchHeader,\n watchMain\n} from \"./components\"\nimport {\n SearchIndex,\n fetchSitemap,\n setupAlternate,\n setupClipboardJS,\n setupInstantNavigation,\n setupVersionSelector\n} from \"./integrations\"\nimport {\n patchEllipsis,\n patchIndeterminate,\n patchScrollfix,\n patchScrolllock\n} from \"./patches\"\nimport \"./polyfills\"\n\n/* ----------------------------------------------------------------------------\n * Functions - @todo refactor\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch search index\n *\n * @returns Search index observable\n */\nfunction fetchSearchIndex(): Observable {\n if (location.protocol === \"file:\") {\n return watchScript(\n `${new URL(\"search/search_index.js\", config.base)}`\n )\n .pipe(\n // @ts-ignore - @todo fix typings\n map(() => __index),\n shareReplay(1)\n )\n } else {\n return requestJSON(\n new URL(\"search/search_index.json\", config.base)\n )\n }\n}\n\n/* ----------------------------------------------------------------------------\n * Application\n * ------------------------------------------------------------------------- */\n\n/* Yay, JavaScript is available */\ndocument.documentElement.classList.remove(\"no-js\")\ndocument.documentElement.classList.add(\"js\")\n\n/* Set up navigation observables and subjects */\nconst document$ = watchDocument()\nconst location$ = watchLocation()\nconst target$ = watchLocationTarget(location$)\nconst keyboard$ = watchKeyboard()\n\n/* Set up media observables */\nconst viewport$ = watchViewport()\nconst tablet$ = watchMedia(\"(min-width: 60em)\")\nconst screen$ = watchMedia(\"(min-width: 76.25em)\")\nconst print$ = watchPrint()\n\n/* Retrieve search index, if search is enabled */\nconst config = configuration()\nconst index$ = document.forms.namedItem(\"search\")\n ? fetchSearchIndex()\n : NEVER\n\n/* Set up Clipboard.js integration */\nconst alert$ = new Subject()\nsetupClipboardJS({ alert$ })\n\n/* Set up language selector */\nsetupAlternate({ document$ })\n\n/* Set up progress indicator */\nconst progress$ = new Subject()\n\n/* Set up sitemap for instant navigation and previews */\nconst sitemap$ = fetchSitemap(config.base)\n\n/* Set up instant navigation, if enabled */\nif (feature(\"navigation.instant\"))\n setupInstantNavigation({ sitemap$, location$, viewport$, progress$ })\n .subscribe(document$)\n\n/* Set up version selector */\nif (config.version?.provider === \"mike\")\n setupVersionSelector({ document$ })\n\n/* Always close drawer and search on navigation */\nmerge(location$, target$)\n .pipe(\n delay(125)\n )\n .subscribe(() => {\n setToggle(\"drawer\", false)\n setToggle(\"search\", false)\n })\n\n/* Set up global keyboard handlers */\nkeyboard$\n .pipe(\n filter(({ mode }) => mode === \"global\")\n )\n .subscribe(key => {\n switch (key.type) {\n\n /* Go to previous page */\n case \"p\":\n case \",\":\n const prev = getOptionalElement(\"link[rel=prev]\")\n if (typeof prev !== \"undefined\")\n setLocation(prev)\n break\n\n /* Go to next page */\n case \"n\":\n case \".\":\n const next = getOptionalElement(\"link[rel=next]\")\n if (typeof next !== \"undefined\")\n setLocation(next)\n break\n\n /* Expand navigation, see https://bit.ly/3ZjG5io */\n case \"Enter\":\n const active = getActiveElement()\n if (active instanceof HTMLLabelElement)\n active.click()\n }\n })\n\n/* Set up patches */\npatchEllipsis({ viewport$, document$ })\npatchIndeterminate({ document$, tablet$ })\npatchScrollfix({ document$ })\npatchScrolllock({ viewport$, tablet$ })\n\n/* Set up header and main area observable */\nconst header$ = watchHeader(getComponentElement(\"header\"), { viewport$ })\nconst main$ = document$\n .pipe(\n map(() => getComponentElement(\"main\")),\n switchMap(el => watchMain(el, { viewport$, header$ })),\n shareReplay(1)\n )\n\n/* Set up control component observables */\nconst control$ = merge(\n\n /* Consent */\n ...getComponentElements(\"consent\")\n .map(el => mountConsent(el, { target$ })),\n\n /* Dialog */\n ...getComponentElements(\"dialog\")\n .map(el => mountDialog(el, { alert$ })),\n\n /* Color palette */\n ...getComponentElements(\"palette\")\n .map(el => mountPalette(el)),\n\n /* Progress bar */\n ...getComponentElements(\"progress\")\n .map(el => mountProgress(el, { progress$ })),\n\n /* Search */\n ...getComponentElements(\"search\")\n .map(el => mountSearch(el, { index$, keyboard$ })),\n\n /* Repository information */\n ...getComponentElements(\"source\")\n .map(el => mountSource(el))\n)\n\n/* Set up content component observables */\nconst content$ = defer(() => merge(\n\n /* Announcement bar */\n ...getComponentElements(\"announce\")\n .map(el => mountAnnounce(el)),\n\n /* Content */\n ...getComponentElements(\"content\")\n .map(el => mountContent(el, { sitemap$, viewport$, target$, print$ })),\n\n /* Search highlighting */\n ...getComponentElements(\"content\")\n .map(el => feature(\"search.highlight\")\n ? mountSearchHiglight(el, { index$, location$ })\n : EMPTY\n ),\n\n /* Header */\n ...getComponentElements(\"header\")\n .map(el => mountHeader(el, { viewport$, header$, main$ })),\n\n /* Header title */\n ...getComponentElements(\"header-title\")\n .map(el => mountHeaderTitle(el, { viewport$, header$ })),\n\n /* Sidebar */\n ...getComponentElements(\"sidebar\")\n .map(el => el.getAttribute(\"data-md-type\") === \"navigation\"\n ? at(screen$, () => mountSidebar(el, { viewport$, header$, main$ }))\n : at(tablet$, () => mountSidebar(el, { viewport$, header$, main$ }))\n ),\n\n /* Navigation tabs */\n ...getComponentElements(\"tabs\")\n .map(el => mountTabs(el, { viewport$, header$ })),\n\n /* Table of contents */\n ...getComponentElements(\"toc\")\n .map(el => mountTableOfContents(el, {\n viewport$, header$, main$, target$\n })),\n\n /* Back-to-top button */\n ...getComponentElements(\"top\")\n .map(el => mountBackToTop(el, { viewport$, header$, main$, target$ }))\n))\n\n/* Set up component observables */\nconst component$ = document$\n .pipe(\n switchMap(() => content$),\n mergeWith(control$),\n shareReplay(1)\n )\n\n/* Subscribe to all components */\ncomponent$.subscribe()\n\n/* ----------------------------------------------------------------------------\n * Exports\n * ------------------------------------------------------------------------- */\n\nwindow.document$ = document$ /* Document observable */\nwindow.location$ = location$ /* Location subject */\nwindow.target$ = target$ /* Location target observable */\nwindow.keyboard$ = keyboard$ /* Keyboard observable */\nwindow.viewport$ = viewport$ /* Viewport observable */\nwindow.tablet$ = tablet$ /* Media tablet observable */\nwindow.screen$ = screen$ /* Media screen observable */\nwindow.print$ = print$ /* Media print observable */\nwindow.alert$ = alert$ /* Alert subject */\nwindow.progress$ = progress$ /* Progress indicator subject */\nwindow.component$ = component$ /* Component observable */\n", "/******************************************************************************\nCopyright (c) Microsoft Corporation.\n\nPermission to use, copy, modify, and/or distribute this software for any\npurpose with or without fee is hereby granted.\n\nTHE SOFTWARE IS PROVIDED \"AS IS\" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH\nREGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY\nAND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT,\nINDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM\nLOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR\nOTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR\nPERFORMANCE OF THIS SOFTWARE.\n***************************************************************************** */\n/* global Reflect, Promise, SuppressedError, Symbol, Iterator */\n\nvar extendStatics = function(d, b) {\n extendStatics = Object.setPrototypeOf ||\n ({ __proto__: [] } instanceof Array && function (d, b) { d.__proto__ = b; }) ||\n function (d, b) { for (var p in b) if (Object.prototype.hasOwnProperty.call(b, p)) d[p] = b[p]; };\n return extendStatics(d, b);\n};\n\nexport function __extends(d, b) {\n if (typeof b !== \"function\" && b !== null)\n throw new TypeError(\"Class extends value \" + String(b) + \" is not a constructor or null\");\n extendStatics(d, b);\n function __() { this.constructor = d; }\n d.prototype = b === null ? Object.create(b) : (__.prototype = b.prototype, new __());\n}\n\nexport var __assign = function() {\n __assign = Object.assign || function __assign(t) {\n for (var s, i = 1, n = arguments.length; i < n; i++) {\n s = arguments[i];\n for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p)) t[p] = s[p];\n }\n return t;\n }\n return __assign.apply(this, arguments);\n}\n\nexport function __rest(s, e) {\n var t = {};\n for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p) && e.indexOf(p) < 0)\n t[p] = s[p];\n if (s != null && typeof Object.getOwnPropertySymbols === \"function\")\n for (var i = 0, p = Object.getOwnPropertySymbols(s); i < p.length; i++) {\n if (e.indexOf(p[i]) < 0 && Object.prototype.propertyIsEnumerable.call(s, p[i]))\n t[p[i]] = s[p[i]];\n }\n return t;\n}\n\nexport function __decorate(decorators, target, key, desc) {\n var c = arguments.length, r = c < 3 ? target : desc === null ? desc = Object.getOwnPropertyDescriptor(target, key) : desc, d;\n if (typeof Reflect === \"object\" && typeof Reflect.decorate === \"function\") r = Reflect.decorate(decorators, target, key, desc);\n else for (var i = decorators.length - 1; i >= 0; i--) if (d = decorators[i]) r = (c < 3 ? d(r) : c > 3 ? d(target, key, r) : d(target, key)) || r;\n return c > 3 && r && Object.defineProperty(target, key, r), r;\n}\n\nexport function __param(paramIndex, decorator) {\n return function (target, key) { decorator(target, key, paramIndex); }\n}\n\nexport function __esDecorate(ctor, descriptorIn, decorators, contextIn, initializers, extraInitializers) {\n function accept(f) { if (f !== void 0 && typeof f !== \"function\") throw new TypeError(\"Function expected\"); return f; }\n var kind = contextIn.kind, key = kind === \"getter\" ? \"get\" : kind === \"setter\" ? \"set\" : \"value\";\n var target = !descriptorIn && ctor ? contextIn[\"static\"] ? ctor : ctor.prototype : null;\n var descriptor = descriptorIn || (target ? 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(function(o, m, k, k2) {\n if (k2 === undefined) k2 = k;\n var desc = Object.getOwnPropertyDescriptor(m, k);\n if (!desc || (\"get\" in desc ? !m.__esModule : desc.writable || desc.configurable)) {\n desc = { enumerable: true, get: function() { return m[k]; } };\n }\n Object.defineProperty(o, k2, desc);\n}) : (function(o, m, k, k2) {\n if (k2 === undefined) k2 = k;\n o[k2] = m[k];\n});\n\nexport function __exportStar(m, o) {\n for (var p in m) if (p !== \"default\" && !Object.prototype.hasOwnProperty.call(o, p)) __createBinding(o, m, p);\n}\n\nexport function __values(o) {\n var s = typeof Symbol === \"function\" && Symbol.iterator, m = s && o[s], i = 0;\n if (m) return m.call(o);\n if (o && typeof o.length === \"number\") return {\n next: function () {\n if (o && i >= o.length) o = void 0;\n return { value: o && o[i++], done: !o };\n }\n };\n throw new TypeError(s ? \"Object is not iterable.\" : \"Symbol.iterator is not defined.\");\n}\n\nexport function __read(o, n) {\n var m = typeof Symbol === \"function\" && o[Symbol.iterator];\n if (!m) return o;\n var i = m.call(o), r, ar = [], e;\n try {\n while ((n === void 0 || n-- > 0) && !(r = i.next()).done) ar.push(r.value);\n }\n catch (error) { e = { error: error }; }\n finally {\n try {\n if (r && !r.done && (m = i[\"return\"])) m.call(i);\n }\n finally { if (e) throw e.error; }\n }\n return ar;\n}\n\n/** @deprecated */\nexport function __spread() {\n for (var ar = [], i = 0; i < arguments.length; i++)\n ar = ar.concat(__read(arguments[i]));\n return ar;\n}\n\n/** @deprecated */\nexport function __spreadArrays() {\n for (var s = 0, i = 0, il = arguments.length; i < il; i++) s += arguments[i].length;\n for (var r = Array(s), k = 0, i = 0; i < il; i++)\n for (var a = arguments[i], j = 0, jl = a.length; j < jl; j++, k++)\n r[k] = a[j];\n return r;\n}\n\nexport function __spreadArray(to, from, pack) {\n if (pack || arguments.length === 2) for (var i = 0, l = from.length, ar; i < l; i++) {\n if (ar || !(i in from)) {\n if (!ar) ar = Array.prototype.slice.call(from, 0, i);\n ar[i] = from[i];\n }\n }\n return to.concat(ar || Array.prototype.slice.call(from));\n}\n\nexport function __await(v) {\n return this instanceof __await ? (this.v = v, this) : new __await(v);\n}\n\nexport function __asyncGenerator(thisArg, _arguments, generator) {\n if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\n var g = generator.apply(thisArg, _arguments || []), i, q = [];\n return i = Object.create((typeof AsyncIterator === \"function\" ? AsyncIterator : Object).prototype), verb(\"next\"), verb(\"throw\"), verb(\"return\", awaitReturn), i[Symbol.asyncIterator] = function () { return this; }, i;\n function awaitReturn(f) { return function (v) { return Promise.resolve(v).then(f, reject); }; }\n function verb(n, f) { if (g[n]) { i[n] = function (v) { return new Promise(function (a, b) { q.push([n, v, a, b]) > 1 || resume(n, v); }); }; if (f) i[n] = f(i[n]); } }\n function resume(n, v) { try { step(g[n](v)); } catch (e) { settle(q[0][3], e); } }\n function step(r) { r.value instanceof __await ? Promise.resolve(r.value.v).then(fulfill, reject) : settle(q[0][2], r); }\n function fulfill(value) { resume(\"next\", value); }\n function reject(value) { resume(\"throw\", value); }\n function settle(f, v) { if (f(v), q.shift(), q.length) resume(q[0][0], q[0][1]); }\n}\n\nexport function __asyncDelegator(o) {\n var i, p;\n return i = {}, verb(\"next\"), verb(\"throw\", function (e) { throw e; }), verb(\"return\"), i[Symbol.iterator] = function () { return this; }, i;\n function verb(n, f) { i[n] = o[n] ? function (v) { return (p = !p) ? { value: __await(o[n](v)), done: false } : f ? f(v) : v; } : f; }\n}\n\nexport function __asyncValues(o) {\n if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\n var m = o[Symbol.asyncIterator], i;\n return m ? m.call(o) : (o = typeof __values === \"function\" ? __values(o) : o[Symbol.iterator](), i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i);\n function verb(n) { i[n] = o[n] && function (v) { return new Promise(function (resolve, reject) { v = o[n](v), settle(resolve, reject, v.done, v.value); }); }; }\n function settle(resolve, reject, d, v) { Promise.resolve(v).then(function(v) { resolve({ value: v, done: d }); }, reject); }\n}\n\nexport function __makeTemplateObject(cooked, raw) {\n if (Object.defineProperty) { Object.defineProperty(cooked, \"raw\", { value: raw }); } else { cooked.raw = raw; }\n return cooked;\n};\n\nvar __setModuleDefault = Object.create ? (function(o, v) {\n Object.defineProperty(o, \"default\", { enumerable: true, value: v });\n}) : function(o, v) {\n o[\"default\"] = v;\n};\n\nexport function __importStar(mod) {\n if (mod && mod.__esModule) return mod;\n var result = {};\n if (mod != null) for (var k in mod) if (k !== \"default\" && Object.prototype.hasOwnProperty.call(mod, k)) __createBinding(result, mod, k);\n __setModuleDefault(result, mod);\n return result;\n}\n\nexport function __importDefault(mod) {\n return (mod && mod.__esModule) ? mod : { default: mod };\n}\n\nexport function __classPrivateFieldGet(receiver, state, kind, f) {\n if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a getter\");\n if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot read private member from an object whose class did not declare it\");\n return kind === \"m\" ? f : kind === \"a\" ? f.call(receiver) : f ? f.value : state.get(receiver);\n}\n\nexport function __classPrivateFieldSet(receiver, state, value, kind, f) {\n if (kind === \"m\") throw new TypeError(\"Private method is not writable\");\n if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a setter\");\n if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot write private member to an object whose class did not declare it\");\n return (kind === \"a\" ? f.call(receiver, value) : f ? f.value = value : state.set(receiver, value)), value;\n}\n\nexport function __classPrivateFieldIn(state, receiver) {\n if (receiver === null || (typeof receiver !== \"object\" && typeof receiver !== \"function\")) throw new TypeError(\"Cannot use 'in' operator on non-object\");\n return typeof state === \"function\" ? receiver === state : state.has(receiver);\n}\n\nexport function __addDisposableResource(env, value, async) {\n if (value !== null && value !== void 0) {\n if (typeof value !== \"object\" && typeof value !== \"function\") throw new TypeError(\"Object expected.\");\n var dispose, inner;\n if (async) {\n if (!Symbol.asyncDispose) throw new TypeError(\"Symbol.asyncDispose is not defined.\");\n dispose = value[Symbol.asyncDispose];\n }\n if (dispose === void 0) {\n if (!Symbol.dispose) throw new TypeError(\"Symbol.dispose is not defined.\");\n dispose = value[Symbol.dispose];\n if (async) inner = dispose;\n }\n if (typeof dispose !== \"function\") throw new TypeError(\"Object not disposable.\");\n if (inner) dispose = function() { try { inner.call(this); } catch (e) { return Promise.reject(e); } };\n env.stack.push({ value: value, dispose: dispose, async: async });\n }\n else if (async) {\n env.stack.push({ async: true });\n }\n return value;\n}\n\nvar _SuppressedError = typeof SuppressedError === \"function\" ? SuppressedError : function (error, suppressed, message) {\n var e = new Error(message);\n return e.name = \"SuppressedError\", e.error = error, e.suppressed = suppressed, e;\n};\n\nexport function __disposeResources(env) {\n function fail(e) {\n env.error = env.hasError ? new _SuppressedError(e, env.error, \"An error was suppressed during disposal.\") : e;\n env.hasError = true;\n }\n var r, s = 0;\n function next() {\n while (r = env.stack.pop()) {\n try {\n if (!r.async && s === 1) return s = 0, env.stack.push(r), Promise.resolve().then(next);\n if (r.dispose) {\n var result = r.dispose.call(r.value);\n if (r.async) return s |= 2, Promise.resolve(result).then(next, function(e) { fail(e); return next(); });\n }\n else s |= 1;\n }\n catch (e) {\n fail(e);\n }\n }\n if (s === 1) return env.hasError ? Promise.reject(env.error) : Promise.resolve();\n if (env.hasError) throw env.error;\n }\n return next();\n}\n\nexport default {\n __extends,\n __assign,\n __rest,\n __decorate,\n __param,\n __metadata,\n __awaiter,\n __generator,\n __createBinding,\n __exportStar,\n __values,\n __read,\n __spread,\n __spreadArrays,\n __spreadArray,\n __await,\n __asyncGenerator,\n __asyncDelegator,\n __asyncValues,\n __makeTemplateObject,\n __importStar,\n __importDefault,\n __classPrivateFieldGet,\n __classPrivateFieldSet,\n __classPrivateFieldIn,\n __addDisposableResource,\n __disposeResources,\n};\n", "/**\n * Returns true if the object is a function.\n * @param value The value to check\n */\nexport function isFunction(value: any): value is (...args: any[]) => any {\n return typeof value === 'function';\n}\n", "/**\n * Used to create Error subclasses until the community moves away from ES5.\n *\n * This is because compiling from TypeScript down to ES5 has issues with subclassing Errors\n * as well as other built-in types: https://github.com/Microsoft/TypeScript/issues/12123\n *\n * @param createImpl A factory function to create the actual constructor implementation. The returned\n * function should be a named function that calls `_super` internally.\n */\nexport function createErrorClass(createImpl: (_super: any) => any): T {\n const _super = (instance: any) => {\n Error.call(instance);\n instance.stack = new Error().stack;\n };\n\n const ctorFunc = createImpl(_super);\n ctorFunc.prototype = Object.create(Error.prototype);\n ctorFunc.prototype.constructor = ctorFunc;\n return ctorFunc;\n}\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface UnsubscriptionError extends Error {\n readonly errors: any[];\n}\n\nexport interface UnsubscriptionErrorCtor {\n /**\n * @deprecated Internal implementation detail. Do not construct error instances.\n * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n */\n new (errors: any[]): UnsubscriptionError;\n}\n\n/**\n * An error thrown when one or more errors have occurred during the\n * `unsubscribe` of a {@link Subscription}.\n */\nexport const UnsubscriptionError: UnsubscriptionErrorCtor = createErrorClass(\n (_super) =>\n function UnsubscriptionErrorImpl(this: any, errors: (Error | string)[]) {\n _super(this);\n this.message = errors\n ? `${errors.length} errors occurred during unsubscription:\n${errors.map((err, i) => `${i + 1}) ${err.toString()}`).join('\\n ')}`\n : '';\n this.name = 'UnsubscriptionError';\n this.errors = errors;\n }\n);\n", "/**\n * Removes an item from an array, mutating it.\n * @param arr The array to remove the item from\n * @param item The item to remove\n */\nexport function arrRemove(arr: T[] | undefined | null, item: T) {\n if (arr) {\n const index = arr.indexOf(item);\n 0 <= index && arr.splice(index, 1);\n }\n}\n", "import { isFunction } from './util/isFunction';\nimport { UnsubscriptionError } from './util/UnsubscriptionError';\nimport { SubscriptionLike, TeardownLogic, Unsubscribable } from './types';\nimport { arrRemove } from './util/arrRemove';\n\n/**\n * Represents a disposable resource, such as the execution of an Observable. A\n * Subscription has one important method, `unsubscribe`, that takes no argument\n * and just disposes the resource held by the subscription.\n *\n * Additionally, subscriptions may be grouped together through the `add()`\n * method, which will attach a child Subscription to the current Subscription.\n * When a Subscription is unsubscribed, all its children (and its grandchildren)\n * will be unsubscribed as well.\n */\nexport class Subscription implements SubscriptionLike {\n public static EMPTY = (() => {\n const empty = new Subscription();\n empty.closed = true;\n return empty;\n })();\n\n /**\n * A flag to indicate whether this Subscription has already been unsubscribed.\n */\n public closed = false;\n\n private _parentage: Subscription[] | Subscription | null = null;\n\n /**\n * The list of registered finalizers to execute upon unsubscription. Adding and removing from this\n * list occurs in the {@link #add} and {@link #remove} methods.\n */\n private _finalizers: Exclude[] | null = null;\n\n /**\n * @param initialTeardown A function executed first as part of the finalization\n * process that is kicked off when {@link #unsubscribe} is called.\n */\n constructor(private initialTeardown?: () => void) {}\n\n /**\n * Disposes the resources held by the subscription. May, for instance, cancel\n * an ongoing Observable execution or cancel any other type of work that\n * started when the Subscription was created.\n */\n unsubscribe(): void {\n let errors: any[] | undefined;\n\n if (!this.closed) {\n this.closed = true;\n\n // Remove this from it's parents.\n const { _parentage } = this;\n if (_parentage) {\n this._parentage = null;\n if (Array.isArray(_parentage)) {\n for (const parent of _parentage) {\n parent.remove(this);\n }\n } else {\n _parentage.remove(this);\n }\n }\n\n const { initialTeardown: initialFinalizer } = this;\n if (isFunction(initialFinalizer)) {\n try {\n initialFinalizer();\n } catch (e) {\n errors = e instanceof UnsubscriptionError ? e.errors : [e];\n }\n }\n\n const { _finalizers } = this;\n if (_finalizers) {\n this._finalizers = null;\n for (const finalizer of _finalizers) {\n try {\n execFinalizer(finalizer);\n } catch (err) {\n errors = errors ?? [];\n if (err instanceof UnsubscriptionError) {\n errors = [...errors, ...err.errors];\n } else {\n errors.push(err);\n }\n }\n }\n }\n\n if (errors) {\n throw new UnsubscriptionError(errors);\n }\n }\n }\n\n /**\n * Adds a finalizer to this subscription, so that finalization will be unsubscribed/called\n * when this subscription is unsubscribed. If this subscription is already {@link #closed},\n * because it has already been unsubscribed, then whatever finalizer is passed to it\n * will automatically be executed (unless the finalizer itself is also a closed subscription).\n *\n * Closed Subscriptions cannot be added as finalizers to any subscription. Adding a closed\n * subscription to a any subscription will result in no operation. (A noop).\n *\n * Adding a subscription to itself, or adding `null` or `undefined` will not perform any\n * operation at all. (A noop).\n *\n * `Subscription` instances that are added to this instance will automatically remove themselves\n * if they are unsubscribed. Functions and {@link Unsubscribable} objects that you wish to remove\n * will need to be removed manually with {@link #remove}\n *\n * @param teardown The finalization logic to add to this subscription.\n */\n add(teardown: TeardownLogic): void {\n // Only add the finalizer if it's not undefined\n // and don't add a subscription to itself.\n if (teardown && teardown !== this) {\n if (this.closed) {\n // If this subscription is already closed,\n // execute whatever finalizer is handed to it automatically.\n execFinalizer(teardown);\n } else {\n if (teardown instanceof Subscription) {\n // We don't add closed subscriptions, and we don't add the same subscription\n // twice. Subscription unsubscribe is idempotent.\n if (teardown.closed || teardown._hasParent(this)) {\n return;\n }\n teardown._addParent(this);\n }\n (this._finalizers = this._finalizers ?? []).push(teardown);\n }\n }\n }\n\n /**\n * Checks to see if a this subscription already has a particular parent.\n * This will signal that this subscription has already been added to the parent in question.\n * @param parent the parent to check for\n */\n private _hasParent(parent: Subscription) {\n const { _parentage } = this;\n return _parentage === parent || (Array.isArray(_parentage) && _parentage.includes(parent));\n }\n\n /**\n * Adds a parent to this subscription so it can be removed from the parent if it\n * unsubscribes on it's own.\n *\n * NOTE: THIS ASSUMES THAT {@link _hasParent} HAS ALREADY BEEN CHECKED.\n * @param parent The parent subscription to add\n */\n private _addParent(parent: Subscription) {\n const { _parentage } = this;\n this._parentage = Array.isArray(_parentage) ? (_parentage.push(parent), _parentage) : _parentage ? [_parentage, parent] : parent;\n }\n\n /**\n * Called on a child when it is removed via {@link #remove}.\n * @param parent The parent to remove\n */\n private _removeParent(parent: Subscription) {\n const { _parentage } = this;\n if (_parentage === parent) {\n this._parentage = null;\n } else if (Array.isArray(_parentage)) {\n arrRemove(_parentage, parent);\n }\n }\n\n /**\n * Removes a finalizer from this subscription that was previously added with the {@link #add} method.\n *\n * Note that `Subscription` instances, when unsubscribed, will automatically remove themselves\n * from every other `Subscription` they have been added to. This means that using the `remove` method\n * is not a common thing and should be used thoughtfully.\n *\n * If you add the same finalizer instance of a function or an unsubscribable object to a `Subscription` instance\n * more than once, you will need to call `remove` the same number of times to remove all instances.\n *\n * All finalizer instances are removed to free up memory upon unsubscription.\n *\n * @param teardown The finalizer to remove from this subscription\n */\n remove(teardown: Exclude): void {\n const { _finalizers } = this;\n _finalizers && arrRemove(_finalizers, teardown);\n\n if (teardown instanceof Subscription) {\n teardown._removeParent(this);\n }\n }\n}\n\nexport const EMPTY_SUBSCRIPTION = Subscription.EMPTY;\n\nexport function isSubscription(value: any): value is Subscription {\n return (\n value instanceof Subscription ||\n (value && 'closed' in value && isFunction(value.remove) && isFunction(value.add) && isFunction(value.unsubscribe))\n );\n}\n\nfunction execFinalizer(finalizer: Unsubscribable | (() => void)) {\n if (isFunction(finalizer)) {\n finalizer();\n } else {\n finalizer.unsubscribe();\n }\n}\n", "import { Subscriber } from './Subscriber';\nimport { ObservableNotification } from './types';\n\n/**\n * The {@link GlobalConfig} object for RxJS. It is used to configure things\n * like how to react on unhandled errors.\n */\nexport const config: GlobalConfig = {\n onUnhandledError: null,\n onStoppedNotification: null,\n Promise: undefined,\n useDeprecatedSynchronousErrorHandling: false,\n useDeprecatedNextContext: false,\n};\n\n/**\n * The global configuration object for RxJS, used to configure things\n * like how to react on unhandled errors. Accessible via {@link config}\n * object.\n */\nexport interface GlobalConfig {\n /**\n * A registration point for unhandled errors from RxJS. These are errors that\n * cannot were not handled by consuming code in the usual subscription path. For\n * example, if you have this configured, and you subscribe to an observable without\n * providing an error handler, errors from that subscription will end up here. This\n * will _always_ be called asynchronously on another job in the runtime. This is because\n * we do not want errors thrown in this user-configured handler to interfere with the\n * behavior of the library.\n */\n onUnhandledError: ((err: any) => void) | null;\n\n /**\n * A registration point for notifications that cannot be sent to subscribers because they\n * have completed, errored or have been explicitly unsubscribed. By default, next, complete\n * and error notifications sent to stopped subscribers are noops. However, sometimes callers\n * might want a different behavior. For example, with sources that attempt to report errors\n * to stopped subscribers, a caller can configure RxJS to throw an unhandled error instead.\n * This will _always_ be called asynchronously on another job in the runtime. This is because\n * we do not want errors thrown in this user-configured handler to interfere with the\n * behavior of the library.\n */\n onStoppedNotification: ((notification: ObservableNotification, subscriber: Subscriber) => void) | null;\n\n /**\n * The promise constructor used by default for {@link Observable#toPromise toPromise} and {@link Observable#forEach forEach}\n * methods.\n *\n * @deprecated As of version 8, RxJS will no longer support this sort of injection of a\n * Promise constructor. If you need a Promise implementation other than native promises,\n * please polyfill/patch Promise as you see appropriate. Will be removed in v8.\n */\n Promise?: PromiseConstructorLike;\n\n /**\n * If true, turns on synchronous error rethrowing, which is a deprecated behavior\n * in v6 and higher. This behavior enables bad patterns like wrapping a subscribe\n * call in a try/catch block. It also enables producer interference, a nasty bug\n * where a multicast can be broken for all observers by a downstream consumer with\n * an unhandled error. DO NOT USE THIS FLAG UNLESS IT'S NEEDED TO BUY TIME\n * FOR MIGRATION REASONS.\n *\n * @deprecated As of version 8, RxJS will no longer support synchronous throwing\n * of unhandled errors. All errors will be thrown on a separate call stack to prevent bad\n * behaviors described above. Will be removed in v8.\n */\n useDeprecatedSynchronousErrorHandling: boolean;\n\n /**\n * If true, enables an as-of-yet undocumented feature from v5: The ability to access\n * `unsubscribe()` via `this` context in `next` functions created in observers passed\n * to `subscribe`.\n *\n * This is being removed because the performance was severely problematic, and it could also cause\n * issues when types other than POJOs are passed to subscribe as subscribers, as they will likely have\n * their `this` context overwritten.\n *\n * @deprecated As of version 8, RxJS will no longer support altering the\n * context of next functions provided as part of an observer to Subscribe. Instead,\n * you will have access to a subscription or a signal or token that will allow you to do things like\n * unsubscribe and test closed status. Will be removed in v8.\n */\n useDeprecatedNextContext: boolean;\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetTimeoutFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearTimeoutFunction = (handle: TimerHandle) => void;\n\ninterface TimeoutProvider {\n setTimeout: SetTimeoutFunction;\n clearTimeout: ClearTimeoutFunction;\n delegate:\n | {\n setTimeout: SetTimeoutFunction;\n clearTimeout: ClearTimeoutFunction;\n }\n | undefined;\n}\n\nexport const timeoutProvider: TimeoutProvider = {\n // When accessing the delegate, use the variable rather than `this` so that\n // the functions can be called without being bound to the provider.\n setTimeout(handler: () => void, timeout?: number, ...args) {\n const { delegate } = timeoutProvider;\n if (delegate?.setTimeout) {\n return delegate.setTimeout(handler, timeout, ...args);\n }\n return setTimeout(handler, timeout, ...args);\n },\n clearTimeout(handle) {\n const { delegate } = timeoutProvider;\n return (delegate?.clearTimeout || clearTimeout)(handle as any);\n },\n delegate: undefined,\n};\n", "import { config } from '../config';\nimport { timeoutProvider } from '../scheduler/timeoutProvider';\n\n/**\n * Handles an error on another job either with the user-configured {@link onUnhandledError},\n * or by throwing it on that new job so it can be picked up by `window.onerror`, `process.on('error')`, etc.\n *\n * This should be called whenever there is an error that is out-of-band with the subscription\n * or when an error hits a terminal boundary of the subscription and no error handler was provided.\n *\n * @param err the error to report\n */\nexport function reportUnhandledError(err: any) {\n timeoutProvider.setTimeout(() => {\n const { onUnhandledError } = config;\n if (onUnhandledError) {\n // Execute the user-configured error handler.\n onUnhandledError(err);\n } else {\n // Throw so it is picked up by the runtime's uncaught error mechanism.\n throw err;\n }\n });\n}\n", "/* tslint:disable:no-empty */\nexport function noop() { }\n", "import { CompleteNotification, NextNotification, ErrorNotification } from './types';\n\n/**\n * A completion object optimized for memory use and created to be the\n * same \"shape\" as other notifications in v8.\n * @internal\n */\nexport const COMPLETE_NOTIFICATION = (() => createNotification('C', undefined, undefined) as CompleteNotification)();\n\n/**\n * Internal use only. Creates an optimized error notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function errorNotification(error: any): ErrorNotification {\n return createNotification('E', undefined, error) as any;\n}\n\n/**\n * Internal use only. Creates an optimized next notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function nextNotification(value: T) {\n return createNotification('N', value, undefined) as NextNotification;\n}\n\n/**\n * Ensures that all notifications created internally have the same \"shape\" in v8.\n *\n * TODO: This is only exported to support a crazy legacy test in `groupBy`.\n * @internal\n */\nexport function createNotification(kind: 'N' | 'E' | 'C', value: any, error: any) {\n return {\n kind,\n value,\n error,\n };\n}\n", "import { config } from '../config';\n\nlet context: { errorThrown: boolean; error: any } | null = null;\n\n/**\n * Handles dealing with errors for super-gross mode. Creates a context, in which\n * any synchronously thrown errors will be passed to {@link captureError}. Which\n * will record the error such that it will be rethrown after the call back is complete.\n * TODO: Remove in v8\n * @param cb An immediately executed function.\n */\nexport function errorContext(cb: () => void) {\n if (config.useDeprecatedSynchronousErrorHandling) {\n const isRoot = !context;\n if (isRoot) {\n context = { errorThrown: false, error: null };\n }\n cb();\n if (isRoot) {\n const { errorThrown, error } = context!;\n context = null;\n if (errorThrown) {\n throw error;\n }\n }\n } else {\n // This is the general non-deprecated path for everyone that\n // isn't crazy enough to use super-gross mode (useDeprecatedSynchronousErrorHandling)\n cb();\n }\n}\n\n/**\n * Captures errors only in super-gross mode.\n * @param err the error to capture\n */\nexport function captureError(err: any) {\n if (config.useDeprecatedSynchronousErrorHandling && context) {\n context.errorThrown = true;\n context.error = err;\n }\n}\n", "import { isFunction } from './util/isFunction';\nimport { Observer, ObservableNotification } from './types';\nimport { isSubscription, Subscription } from './Subscription';\nimport { config } from './config';\nimport { reportUnhandledError } from './util/reportUnhandledError';\nimport { noop } from './util/noop';\nimport { nextNotification, errorNotification, COMPLETE_NOTIFICATION } from './NotificationFactories';\nimport { timeoutProvider } from './scheduler/timeoutProvider';\nimport { captureError } from './util/errorContext';\n\n/**\n * Implements the {@link Observer} interface and extends the\n * {@link Subscription} class. While the {@link Observer} is the public API for\n * consuming the values of an {@link Observable}, all Observers get converted to\n * a Subscriber, in order to provide Subscription-like capabilities such as\n * `unsubscribe`. Subscriber is a common type in RxJS, and crucial for\n * implementing operators, but it is rarely used as a public API.\n */\nexport class Subscriber extends Subscription implements Observer {\n /**\n * A static factory for a Subscriber, given a (potentially partial) definition\n * of an Observer.\n * @param next The `next` callback of an Observer.\n * @param error The `error` callback of an\n * Observer.\n * @param complete The `complete` callback of an\n * Observer.\n * @return A Subscriber wrapping the (partially defined)\n * Observer represented by the given arguments.\n * @deprecated Do not use. Will be removed in v8. There is no replacement for this\n * method, and there is no reason to be creating instances of `Subscriber` directly.\n * If you have a specific use case, please file an issue.\n */\n static create(next?: (x?: T) => void, error?: (e?: any) => void, complete?: () => void): Subscriber {\n return new SafeSubscriber(next, error, complete);\n }\n\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n protected isStopped: boolean = false;\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n protected destination: Subscriber | Observer; // this `any` is the escape hatch to erase extra type param (e.g. R)\n\n /**\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n * There is no reason to directly create an instance of Subscriber. This type is exported for typings reasons.\n */\n constructor(destination?: Subscriber | Observer) {\n super();\n if (destination) {\n this.destination = destination;\n // Automatically chain subscriptions together here.\n // if destination is a Subscription, then it is a Subscriber.\n if (isSubscription(destination)) {\n destination.add(this);\n }\n } else {\n this.destination = EMPTY_OBSERVER;\n }\n }\n\n /**\n * The {@link Observer} callback to receive notifications of type `next` from\n * the Observable, with a value. The Observable may call this method 0 or more\n * times.\n * @param value The `next` value.\n */\n next(value: T): void {\n if (this.isStopped) {\n handleStoppedNotification(nextNotification(value), this);\n } else {\n this._next(value!);\n }\n }\n\n /**\n * The {@link Observer} callback to receive notifications of type `error` from\n * the Observable, with an attached `Error`. Notifies the Observer that\n * the Observable has experienced an error condition.\n * @param err The `error` exception.\n */\n error(err?: any): void {\n if (this.isStopped) {\n handleStoppedNotification(errorNotification(err), this);\n } else {\n this.isStopped = true;\n this._error(err);\n }\n }\n\n /**\n * The {@link Observer} callback to receive a valueless notification of type\n * `complete` from the Observable. Notifies the Observer that the Observable\n * has finished sending push-based notifications.\n */\n complete(): void {\n if (this.isStopped) {\n handleStoppedNotification(COMPLETE_NOTIFICATION, this);\n } else {\n this.isStopped = true;\n this._complete();\n }\n }\n\n unsubscribe(): void {\n if (!this.closed) {\n this.isStopped = true;\n super.unsubscribe();\n this.destination = null!;\n }\n }\n\n protected _next(value: T): void {\n this.destination.next(value);\n }\n\n protected _error(err: any): void {\n try {\n this.destination.error(err);\n } finally {\n this.unsubscribe();\n }\n }\n\n protected _complete(): void {\n try {\n this.destination.complete();\n } finally {\n this.unsubscribe();\n }\n }\n}\n\n/**\n * This bind is captured here because we want to be able to have\n * compatibility with monoid libraries that tend to use a method named\n * `bind`. In particular, a library called Monio requires this.\n */\nconst _bind = Function.prototype.bind;\n\nfunction bind any>(fn: Fn, thisArg: any): Fn {\n return _bind.call(fn, thisArg);\n}\n\n/**\n * Internal optimization only, DO NOT EXPOSE.\n * @internal\n */\nclass ConsumerObserver implements Observer {\n constructor(private partialObserver: Partial>) {}\n\n next(value: T): void {\n const { partialObserver } = this;\n if (partialObserver.next) {\n try {\n partialObserver.next(value);\n } catch (error) {\n handleUnhandledError(error);\n }\n }\n }\n\n error(err: any): void {\n const { partialObserver } = this;\n if (partialObserver.error) {\n try {\n partialObserver.error(err);\n } catch (error) {\n handleUnhandledError(error);\n }\n } else {\n handleUnhandledError(err);\n }\n }\n\n complete(): void {\n const { partialObserver } = this;\n if (partialObserver.complete) {\n try {\n partialObserver.complete();\n } catch (error) {\n handleUnhandledError(error);\n }\n }\n }\n}\n\nexport class SafeSubscriber extends Subscriber {\n constructor(\n observerOrNext?: Partial> | ((value: T) => void) | null,\n error?: ((e?: any) => void) | null,\n complete?: (() => void) | null\n ) {\n super();\n\n let partialObserver: Partial>;\n if (isFunction(observerOrNext) || !observerOrNext) {\n // The first argument is a function, not an observer. The next\n // two arguments *could* be observers, or they could be empty.\n partialObserver = {\n next: (observerOrNext ?? undefined) as ((value: T) => void) | undefined,\n error: error ?? undefined,\n complete: complete ?? undefined,\n };\n } else {\n // The first argument is a partial observer.\n let context: any;\n if (this && config.useDeprecatedNextContext) {\n // This is a deprecated path that made `this.unsubscribe()` available in\n // next handler functions passed to subscribe. This only exists behind a flag\n // now, as it is *very* slow.\n context = Object.create(observerOrNext);\n context.unsubscribe = () => this.unsubscribe();\n partialObserver = {\n next: observerOrNext.next && bind(observerOrNext.next, context),\n error: observerOrNext.error && bind(observerOrNext.error, context),\n complete: observerOrNext.complete && bind(observerOrNext.complete, context),\n };\n } else {\n // The \"normal\" path. Just use the partial observer directly.\n partialObserver = observerOrNext;\n }\n }\n\n // Wrap the partial observer to ensure it's a full observer, and\n // make sure proper error handling is accounted for.\n this.destination = new ConsumerObserver(partialObserver);\n }\n}\n\nfunction handleUnhandledError(error: any) {\n if (config.useDeprecatedSynchronousErrorHandling) {\n captureError(error);\n } else {\n // Ideal path, we report this as an unhandled error,\n // which is thrown on a new call stack.\n reportUnhandledError(error);\n }\n}\n\n/**\n * An error handler used when no error handler was supplied\n * to the SafeSubscriber -- meaning no error handler was supplied\n * do the `subscribe` call on our observable.\n * @param err The error to handle\n */\nfunction defaultErrorHandler(err: any) {\n throw err;\n}\n\n/**\n * A handler for notifications that cannot be sent to a stopped subscriber.\n * @param notification The notification being sent.\n * @param subscriber The stopped subscriber.\n */\nfunction handleStoppedNotification(notification: ObservableNotification, subscriber: Subscriber) {\n const { onStoppedNotification } = config;\n onStoppedNotification && timeoutProvider.setTimeout(() => onStoppedNotification(notification, subscriber));\n}\n\n/**\n * The observer used as a stub for subscriptions where the user did not\n * pass any arguments to `subscribe`. Comes with the default error handling\n * behavior.\n */\nexport const EMPTY_OBSERVER: Readonly> & { closed: true } = {\n closed: true,\n next: noop,\n error: defaultErrorHandler,\n complete: noop,\n};\n", "/**\n * Symbol.observable or a string \"@@observable\". Used for interop\n *\n * @deprecated We will no longer be exporting this symbol in upcoming versions of RxJS.\n * Instead polyfill and use Symbol.observable directly *or* use https://www.npmjs.com/package/symbol-observable\n */\nexport const observable: string | symbol = (() => (typeof Symbol === 'function' && Symbol.observable) || '@@observable')();\n", "/**\n * This function takes one parameter and just returns it. Simply put,\n * this is like `(x: T): T => x`.\n *\n * ## Examples\n *\n * This is useful in some cases when using things like `mergeMap`\n *\n * ```ts\n * import { interval, take, map, range, mergeMap, identity } from 'rxjs';\n *\n * const source$ = interval(1000).pipe(take(5));\n *\n * const result$ = source$.pipe(\n * map(i => range(i)),\n * mergeMap(identity) // same as mergeMap(x => x)\n * );\n *\n * result$.subscribe({\n * next: console.log\n * });\n * ```\n *\n * Or when you want to selectively apply an operator\n *\n * ```ts\n * import { interval, take, identity } from 'rxjs';\n *\n * const shouldLimit = () => Math.random() < 0.5;\n *\n * const source$ = interval(1000);\n *\n * const result$ = source$.pipe(shouldLimit() ? take(5) : identity);\n *\n * result$.subscribe({\n * next: console.log\n * });\n * ```\n *\n * @param x Any value that is returned by this function\n * @returns The value passed as the first parameter to this function\n */\nexport function identity(x: T): T {\n return x;\n}\n", "import { identity } from './identity';\nimport { UnaryFunction } from '../types';\n\nexport function pipe(): typeof identity;\nexport function pipe(fn1: UnaryFunction): UnaryFunction;\nexport function pipe(fn1: UnaryFunction, fn2: UnaryFunction): UnaryFunction;\nexport function pipe(fn1: UnaryFunction, fn2: UnaryFunction, fn3: UnaryFunction): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction,\n fn8: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction,\n fn8: UnaryFunction,\n fn9: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction,\n fn8: UnaryFunction,\n fn9: UnaryFunction,\n ...fns: UnaryFunction[]\n): UnaryFunction;\n\n/**\n * pipe() can be called on one or more functions, each of which can take one argument (\"UnaryFunction\")\n * and uses it to return a value.\n * It returns a function that takes one argument, passes it to the first UnaryFunction, and then\n * passes the result to the next one, passes that result to the next one, and so on. \n */\nexport function pipe(...fns: Array>): UnaryFunction {\n return pipeFromArray(fns);\n}\n\n/** @internal */\nexport function pipeFromArray(fns: Array>): UnaryFunction {\n if (fns.length === 0) {\n return identity as UnaryFunction;\n }\n\n if (fns.length === 1) {\n return fns[0];\n }\n\n return function piped(input: T): R {\n return fns.reduce((prev: any, fn: UnaryFunction) => fn(prev), input as any);\n };\n}\n", "import { Operator } from './Operator';\nimport { SafeSubscriber, Subscriber } from './Subscriber';\nimport { isSubscription, Subscription } from './Subscription';\nimport { TeardownLogic, OperatorFunction, Subscribable, Observer } from './types';\nimport { observable as Symbol_observable } from './symbol/observable';\nimport { pipeFromArray } from './util/pipe';\nimport { config } from './config';\nimport { isFunction } from './util/isFunction';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A representation of any set of values over any amount of time. This is the most basic building block\n * of RxJS.\n */\nexport class Observable implements Subscribable {\n /**\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n */\n source: Observable | undefined;\n\n /**\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n */\n operator: Operator | undefined;\n\n /**\n * @param subscribe The function that is called when the Observable is\n * initially subscribed to. This function is given a Subscriber, to which new values\n * can be `next`ed, or an `error` method can be called to raise an error, or\n * `complete` can be called to notify of a successful completion.\n */\n constructor(subscribe?: (this: Observable, subscriber: Subscriber) => TeardownLogic) {\n if (subscribe) {\n this._subscribe = subscribe;\n }\n }\n\n // HACK: Since TypeScript inherits static properties too, we have to\n // fight against TypeScript here so Subject can have a different static create signature\n /**\n * Creates a new Observable by calling the Observable constructor\n * @param subscribe the subscriber function to be passed to the Observable constructor\n * @return A new observable.\n * @deprecated Use `new Observable()` instead. Will be removed in v8.\n */\n static create: (...args: any[]) => any = (subscribe?: (subscriber: Subscriber) => TeardownLogic) => {\n return new Observable(subscribe);\n };\n\n /**\n * Creates a new Observable, with this Observable instance as the source, and the passed\n * operator defined as the new observable's operator.\n * @param operator the operator defining the operation to take on the observable\n * @return A new observable with the Operator applied.\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n * If you have implemented an operator using `lift`, it is recommended that you create an\n * operator by simply returning `new Observable()` directly. See \"Creating new operators from\n * scratch\" section here: https://rxjs.dev/guide/operators\n */\n lift(operator?: Operator): Observable {\n const observable = new Observable();\n observable.source = this;\n observable.operator = operator;\n return observable;\n }\n\n subscribe(observerOrNext?: Partial> | ((value: T) => void)): Subscription;\n /** @deprecated Instead of passing separate callback arguments, use an observer argument. Signatures taking separate callback arguments will be removed in v8. Details: https://rxjs.dev/deprecations/subscribe-arguments */\n subscribe(next?: ((value: T) => void) | null, error?: ((error: any) => void) | null, complete?: (() => void) | null): Subscription;\n /**\n * Invokes an execution of an Observable and registers Observer handlers for notifications it will emit.\n *\n * Use it when you have all these Observables, but still nothing is happening.\n *\n * `subscribe` is not a regular operator, but a method that calls Observable's internal `subscribe` function. It\n * might be for example a function that you passed to Observable's constructor, but most of the time it is\n * a library implementation, which defines what will be emitted by an Observable, and when it be will emitted. This means\n * that calling `subscribe` is actually the moment when Observable starts its work, not when it is created, as it is often\n * the thought.\n *\n * Apart from starting the execution of an Observable, this method allows you to listen for values\n * that an Observable emits, as well as for when it completes or errors. You can achieve this in two\n * of the following ways.\n *\n * The first way is creating an object that implements {@link Observer} interface. It should have methods\n * defined by that interface, but note that it should be just a regular JavaScript object, which you can create\n * yourself in any way you want (ES6 class, classic function constructor, object literal etc.). In particular, do\n * not attempt to use any RxJS implementation details to create Observers - you don't need them. Remember also\n * that your object does not have to implement all methods. If you find yourself creating a method that doesn't\n * do anything, you can simply omit it. Note however, if the `error` method is not provided and an error happens,\n * it will be thrown asynchronously. Errors thrown asynchronously cannot be caught using `try`/`catch`. Instead,\n * use the {@link onUnhandledError} configuration option or use a runtime handler (like `window.onerror` or\n * `process.on('error)`) to be notified of unhandled errors. Because of this, it's recommended that you provide\n * an `error` method to avoid missing thrown errors.\n *\n * The second way is to give up on Observer object altogether and simply provide callback functions in place of its methods.\n * This means you can provide three functions as arguments to `subscribe`, where the first function is equivalent\n * of a `next` method, the second of an `error` method and the third of a `complete` method. Just as in case of an Observer,\n * if you do not need to listen for something, you can omit a function by passing `undefined` or `null`,\n * since `subscribe` recognizes these functions by where they were placed in function call. When it comes\n * to the `error` function, as with an Observer, if not provided, errors emitted by an Observable will be thrown asynchronously.\n *\n * You can, however, subscribe with no parameters at all. This may be the case where you're not interested in terminal events\n * and you also handled emissions internally by using operators (e.g. using `tap`).\n *\n * Whichever style of calling `subscribe` you use, in both cases it returns a Subscription object.\n * This object allows you to call `unsubscribe` on it, which in turn will stop the work that an Observable does and will clean\n * up all resources that an Observable used. Note that cancelling a subscription will not call `complete` callback\n * provided to `subscribe` function, which is reserved for a regular completion signal that comes from an Observable.\n *\n * Remember that callbacks provided to `subscribe` are not guaranteed to be called asynchronously.\n * It is an Observable itself that decides when these functions will be called. For example {@link of}\n * by default emits all its values synchronously. Always check documentation for how given Observable\n * will behave when subscribed and if its default behavior can be modified with a `scheduler`.\n *\n * #### Examples\n *\n * Subscribe with an {@link guide/observer Observer}\n *\n * ```ts\n * import { of } from 'rxjs';\n *\n * const sumObserver = {\n * sum: 0,\n * next(value) {\n * console.log('Adding: ' + value);\n * this.sum = this.sum + value;\n * },\n * error() {\n * // We actually could just remove this method,\n * // since we do not really care about errors right now.\n * },\n * complete() {\n * console.log('Sum equals: ' + this.sum);\n * }\n * };\n *\n * of(1, 2, 3) // Synchronously emits 1, 2, 3 and then completes.\n * .subscribe(sumObserver);\n *\n * // Logs:\n * // 'Adding: 1'\n * // 'Adding: 2'\n * // 'Adding: 3'\n * // 'Sum equals: 6'\n * ```\n *\n * Subscribe with functions ({@link deprecations/subscribe-arguments deprecated})\n *\n * ```ts\n * import { of } from 'rxjs'\n *\n * let sum = 0;\n *\n * of(1, 2, 3).subscribe(\n * value => {\n * console.log('Adding: ' + value);\n * sum = sum + value;\n * },\n * undefined,\n * () => console.log('Sum equals: ' + sum)\n * );\n *\n * // Logs:\n * // 'Adding: 1'\n * // 'Adding: 2'\n * // 'Adding: 3'\n * // 'Sum equals: 6'\n * ```\n *\n * Cancel a subscription\n *\n * ```ts\n * import { interval } from 'rxjs';\n *\n * const subscription = interval(1000).subscribe({\n * next(num) {\n * console.log(num)\n * },\n * complete() {\n * // Will not be called, even when cancelling subscription.\n * console.log('completed!');\n * }\n * });\n *\n * setTimeout(() => {\n * subscription.unsubscribe();\n * console.log('unsubscribed!');\n * }, 2500);\n *\n * // Logs:\n * // 0 after 1s\n * // 1 after 2s\n * // 'unsubscribed!' after 2.5s\n * ```\n *\n * @param observerOrNext Either an {@link Observer} with some or all callback methods,\n * or the `next` handler that is called for each value emitted from the subscribed Observable.\n * @param error A handler for a terminal event resulting from an error. If no error handler is provided,\n * the error will be thrown asynchronously as unhandled.\n * @param complete A handler for a terminal event resulting from successful completion.\n * @return A subscription reference to the registered handlers.\n */\n subscribe(\n observerOrNext?: Partial> | ((value: T) => void) | null,\n error?: ((error: any) => void) | null,\n complete?: (() => void) | null\n ): Subscription {\n const subscriber = isSubscriber(observerOrNext) ? observerOrNext : new SafeSubscriber(observerOrNext, error, complete);\n\n errorContext(() => {\n const { operator, source } = this;\n subscriber.add(\n operator\n ? // We're dealing with a subscription in the\n // operator chain to one of our lifted operators.\n operator.call(subscriber, source)\n : source\n ? // If `source` has a value, but `operator` does not, something that\n // had intimate knowledge of our API, like our `Subject`, must have\n // set it. We're going to just call `_subscribe` directly.\n this._subscribe(subscriber)\n : // In all other cases, we're likely wrapping a user-provided initializer\n // function, so we need to catch errors and handle them appropriately.\n this._trySubscribe(subscriber)\n );\n });\n\n return subscriber;\n }\n\n /** @internal */\n protected _trySubscribe(sink: Subscriber): TeardownLogic {\n try {\n return this._subscribe(sink);\n } catch (err) {\n // We don't need to return anything in this case,\n // because it's just going to try to `add()` to a subscription\n // above.\n sink.error(err);\n }\n }\n\n /**\n * Used as a NON-CANCELLABLE means of subscribing to an observable, for use with\n * APIs that expect promises, like `async/await`. You cannot unsubscribe from this.\n *\n * **WARNING**: Only use this with observables you *know* will complete. If the source\n * observable does not complete, you will end up with a promise that is hung up, and\n * potentially all of the state of an async function hanging out in memory. To avoid\n * this situation, look into adding something like {@link timeout}, {@link take},\n * {@link takeWhile}, or {@link takeUntil} amongst others.\n *\n * #### Example\n *\n * ```ts\n * import { interval, take } from 'rxjs';\n *\n * const source$ = interval(1000).pipe(take(4));\n *\n * async function getTotal() {\n * let total = 0;\n *\n * await source$.forEach(value => {\n * total += value;\n * console.log('observable -> ' + value);\n * });\n *\n * return total;\n * }\n *\n * getTotal().then(\n * total => console.log('Total: ' + total)\n * );\n *\n * // Expected:\n * // 'observable -> 0'\n * // 'observable -> 1'\n * // 'observable -> 2'\n * // 'observable -> 3'\n * // 'Total: 6'\n * ```\n *\n * @param next A handler for each value emitted by the observable.\n * @return A promise that either resolves on observable completion or\n * rejects with the handled error.\n */\n forEach(next: (value: T) => void): Promise;\n\n /**\n * @param next a handler for each value emitted by the observable\n * @param promiseCtor a constructor function used to instantiate the Promise\n * @return a promise that either resolves on observable completion or\n * rejects with the handled error\n * @deprecated Passing a Promise constructor will no longer be available\n * in upcoming versions of RxJS. This is because it adds weight to the library, for very\n * little benefit. If you need this functionality, it is recommended that you either\n * polyfill Promise, or you create an adapter to convert the returned native promise\n * to whatever promise implementation you wanted. Will be removed in v8.\n */\n forEach(next: (value: T) => void, promiseCtor: PromiseConstructorLike): Promise;\n\n forEach(next: (value: T) => void, promiseCtor?: PromiseConstructorLike): Promise {\n promiseCtor = getPromiseCtor(promiseCtor);\n\n return new promiseCtor((resolve, reject) => {\n const subscriber = new SafeSubscriber({\n next: (value) => {\n try {\n next(value);\n } catch (err) {\n reject(err);\n subscriber.unsubscribe();\n }\n },\n error: reject,\n complete: resolve,\n });\n this.subscribe(subscriber);\n }) as Promise;\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): TeardownLogic {\n return this.source?.subscribe(subscriber);\n }\n\n /**\n * An interop point defined by the es7-observable spec https://github.com/zenparsing/es-observable\n * @return This instance of the observable.\n */\n [Symbol_observable]() {\n return this;\n }\n\n /* tslint:disable:max-line-length */\n pipe(): Observable;\n pipe(op1: OperatorFunction): Observable;\n pipe(op1: OperatorFunction, op2: OperatorFunction): Observable;\n pipe(op1: OperatorFunction, op2: OperatorFunction, op3: OperatorFunction): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction,\n op8: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction,\n op8: OperatorFunction,\n op9: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction,\n op8: OperatorFunction,\n op9: OperatorFunction,\n ...operations: OperatorFunction[]\n ): Observable;\n /* tslint:enable:max-line-length */\n\n /**\n * Used to stitch together functional operators into a chain.\n *\n * ## Example\n *\n * ```ts\n * import { interval, filter, map, scan } from 'rxjs';\n *\n * interval(1000)\n * .pipe(\n * filter(x => x % 2 === 0),\n * map(x => x + x),\n * scan((acc, x) => acc + x)\n * )\n * .subscribe(x => console.log(x));\n * ```\n *\n * @return The Observable result of all the operators having been called\n * in the order they were passed in.\n */\n pipe(...operations: OperatorFunction[]): Observable {\n return pipeFromArray(operations)(this);\n }\n\n /* tslint:disable:max-line-length */\n /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n toPromise(): Promise;\n /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n toPromise(PromiseCtor: typeof Promise): Promise;\n /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n toPromise(PromiseCtor: PromiseConstructorLike): Promise;\n /* tslint:enable:max-line-length */\n\n /**\n * Subscribe to this Observable and get a Promise resolving on\n * `complete` with the last emission (if any).\n *\n * **WARNING**: Only use this with observables you *know* will complete. If the source\n * observable does not complete, you will end up with a promise that is hung up, and\n * potentially all of the state of an async function hanging out in memory. To avoid\n * this situation, look into adding something like {@link timeout}, {@link take},\n * {@link takeWhile}, or {@link takeUntil} amongst others.\n *\n * @param [promiseCtor] a constructor function used to instantiate\n * the Promise\n * @return A Promise that resolves with the last value emit, or\n * rejects on an error. If there were no emissions, Promise\n * resolves with undefined.\n * @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise\n */\n toPromise(promiseCtor?: PromiseConstructorLike): Promise {\n promiseCtor = getPromiseCtor(promiseCtor);\n\n return new promiseCtor((resolve, reject) => {\n let value: T | undefined;\n this.subscribe(\n (x: T) => (value = x),\n (err: any) => reject(err),\n () => resolve(value)\n );\n }) as Promise;\n }\n}\n\n/**\n * Decides between a passed promise constructor from consuming code,\n * A default configured promise constructor, and the native promise\n * constructor and returns it. If nothing can be found, it will throw\n * an error.\n * @param promiseCtor The optional promise constructor to passed by consuming code\n */\nfunction getPromiseCtor(promiseCtor: PromiseConstructorLike | undefined) {\n return promiseCtor ?? config.Promise ?? Promise;\n}\n\nfunction isObserver(value: any): value is Observer {\n return value && isFunction(value.next) && isFunction(value.error) && isFunction(value.complete);\n}\n\nfunction isSubscriber(value: any): value is Subscriber {\n return (value && value instanceof Subscriber) || (isObserver(value) && isSubscription(value));\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { OperatorFunction } from '../types';\nimport { isFunction } from './isFunction';\n\n/**\n * Used to determine if an object is an Observable with a lift function.\n */\nexport function hasLift(source: any): source is { lift: InstanceType['lift'] } {\n return isFunction(source?.lift);\n}\n\n/**\n * Creates an `OperatorFunction`. Used to define operators throughout the library in a concise way.\n * @param init The logic to connect the liftedSource to the subscriber at the moment of subscription.\n */\nexport function operate(\n init: (liftedSource: Observable, subscriber: Subscriber) => (() => void) | void\n): OperatorFunction {\n return (source: Observable) => {\n if (hasLift(source)) {\n return source.lift(function (this: Subscriber, liftedSource: Observable) {\n try {\n return init(liftedSource, this);\n } catch (err) {\n this.error(err);\n }\n });\n }\n throw new TypeError('Unable to lift unknown Observable type');\n };\n}\n", "import { Subscriber } from '../Subscriber';\n\n/**\n * Creates an instance of an `OperatorSubscriber`.\n * @param destination The downstream subscriber.\n * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n * and send to the `destination` error handler.\n * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n * this handler are sent to the `destination` error handler.\n * @param onFinalize Additional teardown logic here. This will only be called on teardown if the\n * subscriber itself is not already closed. This is called after all other teardown logic is executed.\n */\nexport function createOperatorSubscriber(\n destination: Subscriber,\n onNext?: (value: T) => void,\n onComplete?: () => void,\n onError?: (err: any) => void,\n onFinalize?: () => void\n): Subscriber {\n return new OperatorSubscriber(destination, onNext, onComplete, onError, onFinalize);\n}\n\n/**\n * A generic helper for allowing operators to be created with a Subscriber and\n * use closures to capture necessary state from the operator function itself.\n */\nexport class OperatorSubscriber extends Subscriber {\n /**\n * Creates an instance of an `OperatorSubscriber`.\n * @param destination The downstream subscriber.\n * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n * and send to the `destination` error handler.\n * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n * this handler are sent to the `destination` error handler.\n * @param onFinalize Additional finalization logic here. This will only be called on finalization if the\n * subscriber itself is not already closed. This is called after all other finalization logic is executed.\n * @param shouldUnsubscribe An optional check to see if an unsubscribe call should truly unsubscribe.\n * NOTE: This currently **ONLY** exists to support the strange behavior of {@link groupBy}, where unsubscription\n * to the resulting observable does not actually disconnect from the source if there are active subscriptions\n * to any grouped observable. (DO NOT EXPOSE OR USE EXTERNALLY!!!)\n */\n constructor(\n destination: Subscriber,\n onNext?: (value: T) => void,\n onComplete?: () => void,\n onError?: (err: any) => void,\n private onFinalize?: () => void,\n private shouldUnsubscribe?: () => boolean\n ) {\n // It's important - for performance reasons - that all of this class's\n // members are initialized and that they are always initialized in the same\n // order. This will ensure that all OperatorSubscriber instances have the\n // same hidden class in V8. This, in turn, will help keep the number of\n // hidden classes involved in property accesses within the base class as\n // low as possible. If the number of hidden classes involved exceeds four,\n // the property accesses will become megamorphic and performance penalties\n // will be incurred - i.e. inline caches won't be used.\n //\n // The reasons for ensuring all instances have the same hidden class are\n // further discussed in this blog post from Benedikt Meurer:\n // https://benediktmeurer.de/2018/03/23/impact-of-polymorphism-on-component-based-frameworks-like-react/\n super(destination);\n this._next = onNext\n ? function (this: OperatorSubscriber, value: T) {\n try {\n onNext(value);\n } catch (err) {\n destination.error(err);\n }\n }\n : super._next;\n this._error = onError\n ? function (this: OperatorSubscriber, err: any) {\n try {\n onError(err);\n } catch (err) {\n // Send any errors that occur down stream.\n destination.error(err);\n } finally {\n // Ensure finalization.\n this.unsubscribe();\n }\n }\n : super._error;\n this._complete = onComplete\n ? function (this: OperatorSubscriber) {\n try {\n onComplete();\n } catch (err) {\n // Send any errors that occur down stream.\n destination.error(err);\n } finally {\n // Ensure finalization.\n this.unsubscribe();\n }\n }\n : super._complete;\n }\n\n unsubscribe() {\n if (!this.shouldUnsubscribe || this.shouldUnsubscribe()) {\n const { closed } = this;\n super.unsubscribe();\n // Execute additional teardown if we have any and we didn't already do so.\n !closed && this.onFinalize?.();\n }\n }\n}\n", "import { Subscription } from '../Subscription';\n\ninterface AnimationFrameProvider {\n schedule(callback: FrameRequestCallback): Subscription;\n requestAnimationFrame: typeof requestAnimationFrame;\n cancelAnimationFrame: typeof cancelAnimationFrame;\n delegate:\n | {\n requestAnimationFrame: typeof requestAnimationFrame;\n cancelAnimationFrame: typeof cancelAnimationFrame;\n }\n | undefined;\n}\n\nexport const animationFrameProvider: AnimationFrameProvider = {\n // When accessing the delegate, use the variable rather than `this` so that\n // the functions can be called without being bound to the provider.\n schedule(callback) {\n let request = requestAnimationFrame;\n let cancel: typeof cancelAnimationFrame | undefined = cancelAnimationFrame;\n const { delegate } = animationFrameProvider;\n if (delegate) {\n request = delegate.requestAnimationFrame;\n cancel = delegate.cancelAnimationFrame;\n }\n const handle = request((timestamp) => {\n // Clear the cancel function. The request has been fulfilled, so\n // attempting to cancel the request upon unsubscription would be\n // pointless.\n cancel = undefined;\n callback(timestamp);\n });\n return new Subscription(() => cancel?.(handle));\n },\n requestAnimationFrame(...args) {\n const { delegate } = animationFrameProvider;\n return (delegate?.requestAnimationFrame || requestAnimationFrame)(...args);\n },\n cancelAnimationFrame(...args) {\n const { delegate } = animationFrameProvider;\n return (delegate?.cancelAnimationFrame || cancelAnimationFrame)(...args);\n },\n delegate: undefined,\n};\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface ObjectUnsubscribedError extends Error {}\n\nexport interface ObjectUnsubscribedErrorCtor {\n /**\n * @deprecated Internal implementation detail. Do not construct error instances.\n * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n */\n new (): ObjectUnsubscribedError;\n}\n\n/**\n * An error thrown when an action is invalid because the object has been\n * unsubscribed.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n *\n * @class ObjectUnsubscribedError\n */\nexport const ObjectUnsubscribedError: ObjectUnsubscribedErrorCtor = createErrorClass(\n (_super) =>\n function ObjectUnsubscribedErrorImpl(this: any) {\n _super(this);\n this.name = 'ObjectUnsubscribedError';\n this.message = 'object unsubscribed';\n }\n);\n", "import { Operator } from './Operator';\nimport { Observable } from './Observable';\nimport { Subscriber } from './Subscriber';\nimport { Subscription, EMPTY_SUBSCRIPTION } from './Subscription';\nimport { Observer, SubscriptionLike, TeardownLogic } from './types';\nimport { ObjectUnsubscribedError } from './util/ObjectUnsubscribedError';\nimport { arrRemove } from './util/arrRemove';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A Subject is a special type of Observable that allows values to be\n * multicasted to many Observers. Subjects are like EventEmitters.\n *\n * Every Subject is an Observable and an Observer. You can subscribe to a\n * Subject, and you can call next to feed values as well as error and complete.\n */\nexport class Subject extends Observable implements SubscriptionLike {\n closed = false;\n\n private currentObservers: Observer[] | null = null;\n\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n observers: Observer[] = [];\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n isStopped = false;\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n hasError = false;\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n thrownError: any = null;\n\n /**\n * Creates a \"subject\" by basically gluing an observer to an observable.\n *\n * @deprecated Recommended you do not use. Will be removed at some point in the future. Plans for replacement still under discussion.\n */\n static create: (...args: any[]) => any = (destination: Observer, source: Observable): AnonymousSubject => {\n return new AnonymousSubject(destination, source);\n };\n\n constructor() {\n // NOTE: This must be here to obscure Observable's constructor.\n super();\n }\n\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n lift(operator: Operator): Observable {\n const subject = new AnonymousSubject(this, this);\n subject.operator = operator as any;\n return subject as any;\n }\n\n /** @internal */\n protected _throwIfClosed() {\n if (this.closed) {\n throw new ObjectUnsubscribedError();\n }\n }\n\n next(value: T) {\n errorContext(() => {\n this._throwIfClosed();\n if (!this.isStopped) {\n if (!this.currentObservers) {\n this.currentObservers = Array.from(this.observers);\n }\n for (const observer of this.currentObservers) {\n observer.next(value);\n }\n }\n });\n }\n\n error(err: any) {\n errorContext(() => {\n this._throwIfClosed();\n if (!this.isStopped) {\n this.hasError = this.isStopped = true;\n this.thrownError = err;\n const { observers } = this;\n while (observers.length) {\n observers.shift()!.error(err);\n }\n }\n });\n }\n\n complete() {\n errorContext(() => {\n this._throwIfClosed();\n if (!this.isStopped) {\n this.isStopped = true;\n const { observers } = this;\n while (observers.length) {\n observers.shift()!.complete();\n }\n }\n });\n }\n\n unsubscribe() {\n this.isStopped = this.closed = true;\n this.observers = this.currentObservers = null!;\n }\n\n get observed() {\n return this.observers?.length > 0;\n }\n\n /** @internal */\n protected _trySubscribe(subscriber: Subscriber): TeardownLogic {\n this._throwIfClosed();\n return super._trySubscribe(subscriber);\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n this._throwIfClosed();\n this._checkFinalizedStatuses(subscriber);\n return this._innerSubscribe(subscriber);\n }\n\n /** @internal */\n protected _innerSubscribe(subscriber: Subscriber) {\n const { hasError, isStopped, observers } = this;\n if (hasError || isStopped) {\n return EMPTY_SUBSCRIPTION;\n }\n this.currentObservers = null;\n observers.push(subscriber);\n return new Subscription(() => {\n this.currentObservers = null;\n arrRemove(observers, subscriber);\n });\n }\n\n /** @internal */\n protected _checkFinalizedStatuses(subscriber: Subscriber) {\n const { hasError, thrownError, isStopped } = this;\n if (hasError) {\n subscriber.error(thrownError);\n } else if (isStopped) {\n subscriber.complete();\n }\n }\n\n /**\n * Creates a new Observable with this Subject as the source. You can do this\n * to create custom Observer-side logic of the Subject and conceal it from\n * code that uses the Observable.\n * @return Observable that this Subject casts to.\n */\n asObservable(): Observable {\n const observable: any = new Observable();\n observable.source = this;\n return observable;\n }\n}\n\nexport class AnonymousSubject extends Subject {\n constructor(\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n public destination?: Observer,\n source?: Observable\n ) {\n super();\n this.source = source;\n }\n\n next(value: T) {\n this.destination?.next?.(value);\n }\n\n error(err: any) {\n this.destination?.error?.(err);\n }\n\n complete() {\n this.destination?.complete?.();\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n return this.source?.subscribe(subscriber) ?? EMPTY_SUBSCRIPTION;\n }\n}\n", "import { Subject } from './Subject';\nimport { Subscriber } from './Subscriber';\nimport { Subscription } from './Subscription';\n\n/**\n * A variant of Subject that requires an initial value and emits its current\n * value whenever it is subscribed to.\n */\nexport class BehaviorSubject extends Subject {\n constructor(private _value: T) {\n super();\n }\n\n get value(): T {\n return this.getValue();\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n const subscription = super._subscribe(subscriber);\n !subscription.closed && subscriber.next(this._value);\n return subscription;\n }\n\n getValue(): T {\n const { hasError, thrownError, _value } = this;\n if (hasError) {\n throw thrownError;\n }\n this._throwIfClosed();\n return _value;\n }\n\n next(value: T): void {\n super.next((this._value = value));\n }\n}\n", "import { TimestampProvider } from '../types';\n\ninterface DateTimestampProvider extends TimestampProvider {\n delegate: TimestampProvider | undefined;\n}\n\nexport const dateTimestampProvider: DateTimestampProvider = {\n now() {\n // Use the variable rather than `this` so that the function can be called\n // without being bound to the provider.\n return (dateTimestampProvider.delegate || Date).now();\n },\n delegate: undefined,\n};\n", "import { Subject } from './Subject';\nimport { TimestampProvider } from './types';\nimport { Subscriber } from './Subscriber';\nimport { Subscription } from './Subscription';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * A variant of {@link Subject} that \"replays\" old values to new subscribers by emitting them when they first subscribe.\n *\n * `ReplaySubject` has an internal buffer that will store a specified number of values that it has observed. Like `Subject`,\n * `ReplaySubject` \"observes\" values by having them passed to its `next` method. When it observes a value, it will store that\n * value for a time determined by the configuration of the `ReplaySubject`, as passed to its constructor.\n *\n * When a new subscriber subscribes to the `ReplaySubject` instance, it will synchronously emit all values in its buffer in\n * a First-In-First-Out (FIFO) manner. The `ReplaySubject` will also complete, if it has observed completion; and it will\n * error if it has observed an error.\n *\n * There are two main configuration items to be concerned with:\n *\n * 1. `bufferSize` - This will determine how many items are stored in the buffer, defaults to infinite.\n * 2. `windowTime` - The amount of time to hold a value in the buffer before removing it from the buffer.\n *\n * Both configurations may exist simultaneously. So if you would like to buffer a maximum of 3 values, as long as the values\n * are less than 2 seconds old, you could do so with a `new ReplaySubject(3, 2000)`.\n *\n * ### Differences with BehaviorSubject\n *\n * `BehaviorSubject` is similar to `new ReplaySubject(1)`, with a couple of exceptions:\n *\n * 1. `BehaviorSubject` comes \"primed\" with a single value upon construction.\n * 2. `ReplaySubject` will replay values, even after observing an error, where `BehaviorSubject` will not.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n * @see {@link shareReplay}\n */\nexport class ReplaySubject extends Subject {\n private _buffer: (T | number)[] = [];\n private _infiniteTimeWindow = true;\n\n /**\n * @param _bufferSize The size of the buffer to replay on subscription\n * @param _windowTime The amount of time the buffered items will stay buffered\n * @param _timestampProvider An object with a `now()` method that provides the current timestamp. This is used to\n * calculate the amount of time something has been buffered.\n */\n constructor(\n private _bufferSize = Infinity,\n private _windowTime = Infinity,\n private _timestampProvider: TimestampProvider = dateTimestampProvider\n ) {\n super();\n this._infiniteTimeWindow = _windowTime === Infinity;\n this._bufferSize = Math.max(1, _bufferSize);\n this._windowTime = Math.max(1, _windowTime);\n }\n\n next(value: T): void {\n const { isStopped, _buffer, _infiniteTimeWindow, _timestampProvider, _windowTime } = this;\n if (!isStopped) {\n _buffer.push(value);\n !_infiniteTimeWindow && _buffer.push(_timestampProvider.now() + _windowTime);\n }\n this._trimBuffer();\n super.next(value);\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n this._throwIfClosed();\n this._trimBuffer();\n\n const subscription = this._innerSubscribe(subscriber);\n\n const { _infiniteTimeWindow, _buffer } = this;\n // We use a copy here, so reentrant code does not mutate our array while we're\n // emitting it to a new subscriber.\n const copy = _buffer.slice();\n for (let i = 0; i < copy.length && !subscriber.closed; i += _infiniteTimeWindow ? 1 : 2) {\n subscriber.next(copy[i] as T);\n }\n\n this._checkFinalizedStatuses(subscriber);\n\n return subscription;\n }\n\n private _trimBuffer() {\n const { _bufferSize, _timestampProvider, _buffer, _infiniteTimeWindow } = this;\n // If we don't have an infinite buffer size, and we're over the length,\n // use splice to truncate the old buffer values off. Note that we have to\n // double the size for instances where we're not using an infinite time window\n // because we're storing the values and the timestamps in the same array.\n const adjustedBufferSize = (_infiniteTimeWindow ? 1 : 2) * _bufferSize;\n _bufferSize < Infinity && adjustedBufferSize < _buffer.length && _buffer.splice(0, _buffer.length - adjustedBufferSize);\n\n // Now, if we're not in an infinite time window, remove all values where the time is\n // older than what is allowed.\n if (!_infiniteTimeWindow) {\n const now = _timestampProvider.now();\n let last = 0;\n // Search the array for the first timestamp that isn't expired and\n // truncate the buffer up to that point.\n for (let i = 1; i < _buffer.length && (_buffer[i] as number) <= now; i += 2) {\n last = i;\n }\n last && _buffer.splice(0, last + 1);\n }\n }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Subscription } from '../Subscription';\nimport { SchedulerAction } from '../types';\n\n/**\n * A unit of work to be executed in a `scheduler`. An action is typically\n * created from within a {@link SchedulerLike} and an RxJS user does not need to concern\n * themselves about creating and manipulating an Action.\n *\n * ```ts\n * class Action extends Subscription {\n * new (scheduler: Scheduler, work: (state?: T) => void);\n * schedule(state?: T, delay: number = 0): Subscription;\n * }\n * ```\n */\nexport class Action extends Subscription {\n constructor(scheduler: Scheduler, work: (this: SchedulerAction, state?: T) => void) {\n super();\n }\n /**\n * Schedules this action on its parent {@link SchedulerLike} for execution. May be passed\n * some context object, `state`. May happen at some point in the future,\n * according to the `delay` parameter, if specified.\n * @param state Some contextual data that the `work` function uses when called by the\n * Scheduler.\n * @param delay Time to wait before executing the work, where the time unit is implicit\n * and defined by the Scheduler.\n * @return A subscription in order to be able to unsubscribe the scheduled work.\n */\n public schedule(state?: T, delay: number = 0): Subscription {\n return this;\n }\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetIntervalFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearIntervalFunction = (handle: TimerHandle) => void;\n\ninterface IntervalProvider {\n setInterval: SetIntervalFunction;\n clearInterval: ClearIntervalFunction;\n delegate:\n | {\n setInterval: SetIntervalFunction;\n clearInterval: ClearIntervalFunction;\n }\n | undefined;\n}\n\nexport const intervalProvider: IntervalProvider = {\n // When accessing the delegate, use the variable rather than `this` so that\n // the functions can be called without being bound to the provider.\n setInterval(handler: () => void, timeout?: number, ...args) {\n const { delegate } = intervalProvider;\n if (delegate?.setInterval) {\n return delegate.setInterval(handler, timeout, ...args);\n }\n return setInterval(handler, timeout, ...args);\n },\n clearInterval(handle) {\n const { delegate } = intervalProvider;\n return (delegate?.clearInterval || clearInterval)(handle as any);\n },\n delegate: undefined,\n};\n", "import { Action } from './Action';\nimport { SchedulerAction } from '../types';\nimport { Subscription } from '../Subscription';\nimport { AsyncScheduler } from './AsyncScheduler';\nimport { intervalProvider } from './intervalProvider';\nimport { arrRemove } from '../util/arrRemove';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncAction extends Action {\n public id: TimerHandle | undefined;\n public state?: T;\n // @ts-ignore: Property has no initializer and is not definitely assigned\n public delay: number;\n protected pending: boolean = false;\n\n constructor(protected scheduler: AsyncScheduler, protected work: (this: SchedulerAction, state?: T) => void) {\n super(scheduler, work);\n }\n\n public schedule(state?: T, delay: number = 0): Subscription {\n if (this.closed) {\n return this;\n }\n\n // Always replace the current state with the new state.\n this.state = state;\n\n const id = this.id;\n const scheduler = this.scheduler;\n\n //\n // Important implementation note:\n //\n // Actions only execute once by default, unless rescheduled from within the\n // scheduled callback. This allows us to implement single and repeat\n // actions via the same code path, without adding API surface area, as well\n // as mimic traditional recursion but across asynchronous boundaries.\n //\n // However, JS runtimes and timers distinguish between intervals achieved by\n // serial `setTimeout` calls vs. a single `setInterval` call. An interval of\n // serial `setTimeout` calls can be individually delayed, which delays\n // scheduling the next `setTimeout`, and so on. `setInterval` attempts to\n // guarantee the interval callback will be invoked more precisely to the\n // interval period, regardless of load.\n //\n // Therefore, we use `setInterval` to schedule single and repeat actions.\n // If the action reschedules itself with the same delay, the interval is not\n // canceled. If the action doesn't reschedule, or reschedules with a\n // different delay, the interval will be canceled after scheduled callback\n // execution.\n //\n if (id != null) {\n this.id = this.recycleAsyncId(scheduler, id, delay);\n }\n\n // Set the pending flag indicating that this action has been scheduled, or\n // has recursively rescheduled itself.\n this.pending = true;\n\n this.delay = delay;\n // If this action has already an async Id, don't request a new one.\n this.id = this.id ?? this.requestAsyncId(scheduler, this.id, delay);\n\n return this;\n }\n\n protected requestAsyncId(scheduler: AsyncScheduler, _id?: TimerHandle, delay: number = 0): TimerHandle {\n return intervalProvider.setInterval(scheduler.flush.bind(scheduler, this), delay);\n }\n\n protected recycleAsyncId(_scheduler: AsyncScheduler, id?: TimerHandle, delay: number | null = 0): TimerHandle | undefined {\n // If this action is rescheduled with the same delay time, don't clear the interval id.\n if (delay != null && this.delay === delay && this.pending === false) {\n return id;\n }\n // Otherwise, if the action's delay time is different from the current delay,\n // or the action has been rescheduled before it's executed, clear the interval id\n if (id != null) {\n intervalProvider.clearInterval(id);\n }\n\n return undefined;\n }\n\n /**\n * Immediately executes this action and the `work` it contains.\n */\n public execute(state: T, delay: number): any {\n if (this.closed) {\n return new Error('executing a cancelled action');\n }\n\n this.pending = false;\n const error = this._execute(state, delay);\n if (error) {\n return error;\n } else if (this.pending === false && this.id != null) {\n // Dequeue if the action didn't reschedule itself. Don't call\n // unsubscribe(), because the action could reschedule later.\n // For example:\n // ```\n // scheduler.schedule(function doWork(counter) {\n // /* ... I'm a busy worker bee ... */\n // var originalAction = this;\n // /* wait 100ms before rescheduling the action */\n // setTimeout(function () {\n // originalAction.schedule(counter + 1);\n // }, 100);\n // }, 1000);\n // ```\n this.id = this.recycleAsyncId(this.scheduler, this.id, null);\n }\n }\n\n protected _execute(state: T, _delay: number): any {\n let errored: boolean = false;\n let errorValue: any;\n try {\n this.work(state);\n } catch (e) {\n errored = true;\n // HACK: Since code elsewhere is relying on the \"truthiness\" of the\n // return here, we can't have it return \"\" or 0 or false.\n // TODO: Clean this up when we refactor schedulers mid-version-8 or so.\n errorValue = e ? e : new Error('Scheduled action threw falsy error');\n }\n if (errored) {\n this.unsubscribe();\n return errorValue;\n }\n }\n\n unsubscribe() {\n if (!this.closed) {\n const { id, scheduler } = this;\n const { actions } = scheduler;\n\n this.work = this.state = this.scheduler = null!;\n this.pending = false;\n\n arrRemove(actions, this);\n if (id != null) {\n this.id = this.recycleAsyncId(scheduler, id, null);\n }\n\n this.delay = null!;\n super.unsubscribe();\n }\n }\n}\n", "import { Action } from './scheduler/Action';\nimport { Subscription } from './Subscription';\nimport { SchedulerLike, SchedulerAction } from './types';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * An execution context and a data structure to order tasks and schedule their\n * execution. Provides a notion of (potentially virtual) time, through the\n * `now()` getter method.\n *\n * Each unit of work in a Scheduler is called an `Action`.\n *\n * ```ts\n * class Scheduler {\n * now(): number;\n * schedule(work, delay?, state?): Subscription;\n * }\n * ```\n *\n * @deprecated Scheduler is an internal implementation detail of RxJS, and\n * should not be used directly. Rather, create your own class and implement\n * {@link SchedulerLike}. Will be made internal in v8.\n */\nexport class Scheduler implements SchedulerLike {\n public static now: () => number = dateTimestampProvider.now;\n\n constructor(private schedulerActionCtor: typeof Action, now: () => number = Scheduler.now) {\n this.now = now;\n }\n\n /**\n * A getter method that returns a number representing the current time\n * (at the time this function was called) according to the scheduler's own\n * internal clock.\n * @return A number that represents the current time. May or may not\n * have a relation to wall-clock time. May or may not refer to a time unit\n * (e.g. milliseconds).\n */\n public now: () => number;\n\n /**\n * Schedules a function, `work`, for execution. May happen at some point in\n * the future, according to the `delay` parameter, if specified. May be passed\n * some context object, `state`, which will be passed to the `work` function.\n *\n * The given arguments will be processed an stored as an Action object in a\n * queue of actions.\n *\n * @param work A function representing a task, or some unit of work to be\n * executed by the Scheduler.\n * @param delay Time to wait before executing the work, where the time unit is\n * implicit and defined by the Scheduler itself.\n * @param state Some contextual data that the `work` function uses when called\n * by the Scheduler.\n * @return A subscription in order to be able to unsubscribe the scheduled work.\n */\n public schedule(work: (this: SchedulerAction, state?: T) => void, delay: number = 0, state?: T): Subscription {\n return new this.schedulerActionCtor(this, work).schedule(state, delay);\n }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Action } from './Action';\nimport { AsyncAction } from './AsyncAction';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncScheduler extends Scheduler {\n public actions: Array> = [];\n /**\n * A flag to indicate whether the Scheduler is currently executing a batch of\n * queued actions.\n * @internal\n */\n public _active: boolean = false;\n /**\n * An internal ID used to track the latest asynchronous task such as those\n * coming from `setTimeout`, `setInterval`, `requestAnimationFrame`, and\n * others.\n * @internal\n */\n public _scheduled: TimerHandle | undefined;\n\n constructor(SchedulerAction: typeof Action, now: () => number = Scheduler.now) {\n super(SchedulerAction, now);\n }\n\n public flush(action: AsyncAction): void {\n const { actions } = this;\n\n if (this._active) {\n actions.push(action);\n return;\n }\n\n let error: any;\n this._active = true;\n\n do {\n if ((error = action.execute(action.state, action.delay))) {\n break;\n }\n } while ((action = actions.shift()!)); // exhaust the scheduler queue\n\n this._active = false;\n\n if (error) {\n while ((action = actions.shift()!)) {\n action.unsubscribe();\n }\n throw error;\n }\n }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\n/**\n *\n * Async Scheduler\n *\n * Schedule task as if you used setTimeout(task, duration)\n *\n * `async` scheduler schedules tasks asynchronously, by putting them on the JavaScript\n * event loop queue. It is best used to delay tasks in time or to schedule tasks repeating\n * in intervals.\n *\n * If you just want to \"defer\" task, that is to perform it right after currently\n * executing synchronous code ends (commonly achieved by `setTimeout(deferredTask, 0)`),\n * better choice will be the {@link asapScheduler} scheduler.\n *\n * ## Examples\n * Use async scheduler to delay task\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * const task = () => console.log('it works!');\n *\n * asyncScheduler.schedule(task, 2000);\n *\n * // After 2 seconds logs:\n * // \"it works!\"\n * ```\n *\n * Use async scheduler to repeat task in intervals\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * function task(state) {\n * console.log(state);\n * this.schedule(state + 1, 1000); // `this` references currently executing Action,\n * // which we reschedule with new state and delay\n * }\n *\n * asyncScheduler.schedule(task, 3000, 0);\n *\n * // Logs:\n * // 0 after 3s\n * // 1 after 4s\n * // 2 after 5s\n * // 3 after 6s\n * ```\n */\n\nexport const asyncScheduler = new AsyncScheduler(AsyncAction);\n\n/**\n * @deprecated Renamed to {@link asyncScheduler}. Will be removed in v8.\n */\nexport const async = asyncScheduler;\n", "import { AsyncAction } from './AsyncAction';\nimport { Subscription } from '../Subscription';\nimport { QueueScheduler } from './QueueScheduler';\nimport { SchedulerAction } from '../types';\nimport { TimerHandle } from './timerHandle';\n\nexport class QueueAction extends AsyncAction {\n constructor(protected scheduler: QueueScheduler, protected work: (this: SchedulerAction, state?: T) => void) {\n super(scheduler, work);\n }\n\n public schedule(state?: T, delay: number = 0): Subscription {\n if (delay > 0) {\n return super.schedule(state, delay);\n }\n this.delay = delay;\n this.state = state;\n this.scheduler.flush(this);\n return this;\n }\n\n public execute(state: T, delay: number): any {\n return delay > 0 || this.closed ? super.execute(state, delay) : this._execute(state, delay);\n }\n\n protected requestAsyncId(scheduler: QueueScheduler, id?: TimerHandle, delay: number = 0): TimerHandle {\n // If delay exists and is greater than 0, or if the delay is null (the\n // action wasn't rescheduled) but was originally scheduled as an async\n // action, then recycle as an async action.\n\n if ((delay != null && delay > 0) || (delay == null && this.delay > 0)) {\n return super.requestAsyncId(scheduler, id, delay);\n }\n\n // Otherwise flush the scheduler starting with this action.\n scheduler.flush(this);\n\n // HACK: In the past, this was returning `void`. However, `void` isn't a valid\n // `TimerHandle`, and generally the return value here isn't really used. So the\n // compromise is to return `0` which is both \"falsy\" and a valid `TimerHandle`,\n // as opposed to refactoring every other instanceo of `requestAsyncId`.\n return 0;\n }\n}\n", "import { AsyncScheduler } from './AsyncScheduler';\n\nexport class QueueScheduler extends AsyncScheduler {\n}\n", "import { QueueAction } from './QueueAction';\nimport { QueueScheduler } from './QueueScheduler';\n\n/**\n *\n * Queue Scheduler\n *\n * Put every next task on a queue, instead of executing it immediately\n *\n * `queue` scheduler, when used with delay, behaves the same as {@link asyncScheduler} scheduler.\n *\n * When used without delay, it schedules given task synchronously - executes it right when\n * it is scheduled. However when called recursively, that is when inside the scheduled task,\n * another task is scheduled with queue scheduler, instead of executing immediately as well,\n * that task will be put on a queue and wait for current one to finish.\n *\n * This means that when you execute task with `queue` scheduler, you are sure it will end\n * before any other task scheduled with that scheduler will start.\n *\n * ## Examples\n * Schedule recursively first, then do something\n * ```ts\n * import { queueScheduler } from 'rxjs';\n *\n * queueScheduler.schedule(() => {\n * queueScheduler.schedule(() => console.log('second')); // will not happen now, but will be put on a queue\n *\n * console.log('first');\n * });\n *\n * // Logs:\n * // \"first\"\n * // \"second\"\n * ```\n *\n * Reschedule itself recursively\n * ```ts\n * import { queueScheduler } from 'rxjs';\n *\n * queueScheduler.schedule(function(state) {\n * if (state !== 0) {\n * console.log('before', state);\n * this.schedule(state - 1); // `this` references currently executing Action,\n * // which we reschedule with new state\n * console.log('after', state);\n * }\n * }, 0, 3);\n *\n * // In scheduler that runs recursively, you would expect:\n * // \"before\", 3\n * // \"before\", 2\n * // \"before\", 1\n * // \"after\", 1\n * // \"after\", 2\n * // \"after\", 3\n *\n * // But with queue it logs:\n * // \"before\", 3\n * // \"after\", 3\n * // \"before\", 2\n * // \"after\", 2\n * // \"before\", 1\n * // \"after\", 1\n * ```\n */\n\nexport const queueScheduler = new QueueScheduler(QueueAction);\n\n/**\n * @deprecated Renamed to {@link queueScheduler}. Will be removed in v8.\n */\nexport const queue = queueScheduler;\n", "import { AsyncAction } from './AsyncAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\nimport { SchedulerAction } from '../types';\nimport { animationFrameProvider } from './animationFrameProvider';\nimport { TimerHandle } from './timerHandle';\n\nexport class AnimationFrameAction extends AsyncAction {\n constructor(protected scheduler: AnimationFrameScheduler, protected work: (this: SchedulerAction, state?: T) => void) {\n super(scheduler, work);\n }\n\n protected requestAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle {\n // If delay is greater than 0, request as an async action.\n if (delay !== null && delay > 0) {\n return super.requestAsyncId(scheduler, id, delay);\n }\n // Push the action to the end of the scheduler queue.\n scheduler.actions.push(this);\n // If an animation frame has already been requested, don't request another\n // one. If an animation frame hasn't been requested yet, request one. Return\n // the current animation frame request id.\n return scheduler._scheduled || (scheduler._scheduled = animationFrameProvider.requestAnimationFrame(() => scheduler.flush(undefined)));\n }\n\n protected recycleAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle | undefined {\n // If delay exists and is greater than 0, or if the delay is null (the\n // action wasn't rescheduled) but was originally scheduled as an async\n // action, then recycle as an async action.\n if (delay != null ? delay > 0 : this.delay > 0) {\n return super.recycleAsyncId(scheduler, id, delay);\n }\n // If the scheduler queue has no remaining actions with the same async id,\n // cancel the requested animation frame and set the scheduled flag to\n // undefined so the next AnimationFrameAction will request its own.\n const { actions } = scheduler;\n if (id != null && id === scheduler._scheduled && actions[actions.length - 1]?.id !== id) {\n animationFrameProvider.cancelAnimationFrame(id as number);\n scheduler._scheduled = undefined;\n }\n // Return undefined so the action knows to request a new async id if it's rescheduled.\n return undefined;\n }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\nexport class AnimationFrameScheduler extends AsyncScheduler {\n public flush(action?: AsyncAction): void {\n this._active = true;\n // The async id that effects a call to flush is stored in _scheduled.\n // Before executing an action, it's necessary to check the action's async\n // id to determine whether it's supposed to be executed in the current\n // flush.\n // Previous implementations of this method used a count to determine this,\n // but that was unsound, as actions that are unsubscribed - i.e. cancelled -\n // are removed from the actions array and that can shift actions that are\n // scheduled to be executed in a subsequent flush into positions at which\n // they are executed within the current flush.\n let flushId;\n if (action) {\n flushId = action.id;\n } else {\n flushId = this._scheduled;\n this._scheduled = undefined;\n }\n\n const { actions } = this;\n let error: any;\n action = action || actions.shift()!;\n\n do {\n if ((error = action.execute(action.state, action.delay))) {\n break;\n }\n } while ((action = actions[0]) && action.id === flushId && actions.shift());\n\n this._active = false;\n\n if (error) {\n while ((action = actions[0]) && action.id === flushId && actions.shift()) {\n action.unsubscribe();\n }\n throw error;\n }\n }\n}\n", "import { AnimationFrameAction } from './AnimationFrameAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\n\n/**\n *\n * Animation Frame Scheduler\n *\n * Perform task when `window.requestAnimationFrame` would fire\n *\n * When `animationFrame` scheduler is used with delay, it will fall back to {@link asyncScheduler} scheduler\n * behaviour.\n *\n * Without delay, `animationFrame` scheduler can be used to create smooth browser animations.\n * It makes sure scheduled task will happen just before next browser content repaint,\n * thus performing animations as efficiently as possible.\n *\n * ## Example\n * Schedule div height animation\n * ```ts\n * // html:
\n * import { animationFrameScheduler } from 'rxjs';\n *\n * const div = document.querySelector('div');\n *\n * animationFrameScheduler.schedule(function(height) {\n * div.style.height = height + \"px\";\n *\n * this.schedule(height + 1); // `this` references currently executing Action,\n * // which we reschedule with new state\n * }, 0, 0);\n *\n * // You will see a div element growing in height\n * ```\n */\n\nexport const animationFrameScheduler = new AnimationFrameScheduler(AnimationFrameAction);\n\n/**\n * @deprecated Renamed to {@link animationFrameScheduler}. Will be removed in v8.\n */\nexport const animationFrame = animationFrameScheduler;\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\n\n/**\n * A simple Observable that emits no items to the Observer and immediately\n * emits a complete notification.\n *\n * Just emits 'complete', and nothing else.\n *\n * ![](empty.png)\n *\n * A simple Observable that only emits the complete notification. It can be used\n * for composing with other Observables, such as in a {@link mergeMap}.\n *\n * ## Examples\n *\n * Log complete notification\n *\n * ```ts\n * import { EMPTY } from 'rxjs';\n *\n * EMPTY.subscribe({\n * next: () => console.log('Next'),\n * complete: () => console.log('Complete!')\n * });\n *\n * // Outputs\n * // Complete!\n * ```\n *\n * Emit the number 7, then complete\n *\n * ```ts\n * import { EMPTY, startWith } from 'rxjs';\n *\n * const result = EMPTY.pipe(startWith(7));\n * result.subscribe(x => console.log(x));\n *\n * // Outputs\n * // 7\n * ```\n *\n * Map and flatten only odd numbers to the sequence `'a'`, `'b'`, `'c'`\n *\n * ```ts\n * import { interval, mergeMap, of, EMPTY } from 'rxjs';\n *\n * const interval$ = interval(1000);\n * const result = interval$.pipe(\n * mergeMap(x => x % 2 === 1 ? of('a', 'b', 'c') : EMPTY),\n * );\n * result.subscribe(x => console.log(x));\n *\n * // Results in the following to the console:\n * // x is equal to the count on the interval, e.g. (0, 1, 2, 3, ...)\n * // x will occur every 1000ms\n * // if x % 2 is equal to 1, print a, b, c (each on its own)\n * // if x % 2 is not equal to 1, nothing will be output\n * ```\n *\n * @see {@link Observable}\n * @see {@link NEVER}\n * @see {@link of}\n * @see {@link throwError}\n */\nexport const EMPTY = new Observable((subscriber) => subscriber.complete());\n\n/**\n * @param scheduler A {@link SchedulerLike} to use for scheduling\n * the emission of the complete notification.\n * @deprecated Replaced with the {@link EMPTY} constant or {@link scheduled} (e.g. `scheduled([], scheduler)`). Will be removed in v8.\n */\nexport function empty(scheduler?: SchedulerLike) {\n return scheduler ? emptyScheduled(scheduler) : EMPTY;\n}\n\nfunction emptyScheduled(scheduler: SchedulerLike) {\n return new Observable((subscriber) => scheduler.schedule(() => subscriber.complete()));\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport function isScheduler(value: any): value is SchedulerLike {\n return value && isFunction(value.schedule);\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\nimport { isScheduler } from './isScheduler';\n\nfunction last(arr: T[]): T | undefined {\n return arr[arr.length - 1];\n}\n\nexport function popResultSelector(args: any[]): ((...args: unknown[]) => unknown) | undefined {\n return isFunction(last(args)) ? args.pop() : undefined;\n}\n\nexport function popScheduler(args: any[]): SchedulerLike | undefined {\n return isScheduler(last(args)) ? args.pop() : undefined;\n}\n\nexport function popNumber(args: any[], defaultValue: number): number {\n return typeof last(args) === 'number' ? args.pop()! : defaultValue;\n}\n", "export const isArrayLike = ((x: any): x is ArrayLike => x && typeof x.length === 'number' && typeof x !== 'function');", "import { isFunction } from \"./isFunction\";\n\n/**\n * Tests to see if the object is \"thennable\".\n * @param value the object to test\n */\nexport function isPromise(value: any): value is PromiseLike {\n return isFunction(value?.then);\n}\n", "import { InteropObservable } from '../types';\nimport { observable as Symbol_observable } from '../symbol/observable';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being Observable (but not necessary an Rx Observable) */\nexport function isInteropObservable(input: any): input is InteropObservable {\n return isFunction(input[Symbol_observable]);\n}\n", "import { isFunction } from './isFunction';\n\nexport function isAsyncIterable(obj: any): obj is AsyncIterable {\n return Symbol.asyncIterator && isFunction(obj?.[Symbol.asyncIterator]);\n}\n", "/**\n * Creates the TypeError to throw if an invalid object is passed to `from` or `scheduled`.\n * @param input The object that was passed.\n */\nexport function createInvalidObservableTypeError(input: any) {\n // TODO: We should create error codes that can be looked up, so this can be less verbose.\n return new TypeError(\n `You provided ${\n input !== null && typeof input === 'object' ? 'an invalid object' : `'${input}'`\n } where a stream was expected. You can provide an Observable, Promise, ReadableStream, Array, AsyncIterable, or Iterable.`\n );\n}\n", "export function getSymbolIterator(): symbol {\n if (typeof Symbol !== 'function' || !Symbol.iterator) {\n return '@@iterator' as any;\n }\n\n return Symbol.iterator;\n}\n\nexport const iterator = getSymbolIterator();\n", "import { iterator as Symbol_iterator } from '../symbol/iterator';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being an Iterable */\nexport function isIterable(input: any): input is Iterable {\n return isFunction(input?.[Symbol_iterator]);\n}\n", "import { ReadableStreamLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport async function* readableStreamLikeToAsyncGenerator(readableStream: ReadableStreamLike): AsyncGenerator {\n const reader = readableStream.getReader();\n try {\n while (true) {\n const { value, done } = await reader.read();\n if (done) {\n return;\n }\n yield value!;\n }\n } finally {\n reader.releaseLock();\n }\n}\n\nexport function isReadableStreamLike(obj: any): obj is ReadableStreamLike {\n // We don't want to use instanceof checks because they would return\n // false for instances from another Realm, like an